Polynomials
NOTES: A quick overview of polynomials You might want to see my overview of algebraic expressions first to go over using the distributive property (including distributing a negative) and combining like terms.
What is a polynomial?
It’s a type of algebraic expression (one with no variables in the denominator or square roots).
It involves one or more terms combined by addition or subtraction. o If there’s subtraction, the term after it is considered negative.
The polynomial 3x2 + 4x 5 contains 3 terms: 3x2, 4x, and 5. This may be easier to see if you rewrite subtraction as the addition of a negative: 3x2 + 4x + 5
Polynomials are named by how many terms they have: Examples 2 2 monomial: 1 term 3y 𝑧 5a2b3 binomial: 2 terms trinomial: 3 terms
2x 4.5y 6x +
3
5y2 + 3z
2
𝑦 3z 5
3a2 + 2bc
4x2 + 3y2 xy
How do you evaluate polynomials?
2a2b + 3ab2 5ab
“Plug ’n play.”
(To evaluate an expression means to find its value, given the values of the variables.)
1. First plug in all the values on one line to replace the variables. o Use parentheses around the values you substitute. This will indicate multiplication and mark negative numbers. 2. Then crank out the rest with the order of operations, doing one step per line. EXAMPLES: Evaluate 6x – 3y + xy for x = 4 and y = 2
Evaluate 3x2 – 2y2 + xy for x = 5 and y = –3
6x – 3y + xy = 6(4) – 3(2) + (4)(2) = 24 – 6 + 8 = 26
3x2 – 2y2 + xy = 3(5)2 – 2(–3)2 + (5)(–3) = 3(25) – 2(9) + (5)(–3) = 75 – 18 + (–15) = 57 + (–15) = 42
© D. Stark 2017
10/3/2017
OVERVIEW: polynomials
1
How do you add polynomials? 1. Drop any parentheses. They don’t matter for addition. 2. Combine like terms. Remember to “Take what’s left”—take along any symbol on the left of a term as part of that term. 3. If necessary, clean things up for a fully simplified, standard form answer.
EXAMPLES: (2x2 + 4x) + (3x2 7x) 2x2
+ 4x
+ 3x2
– 7x
ANSWER: 5x2 3x
(–4xy2 + 2x 8) + (3xy2 x) 8
–4xy2 + 2x
+ 3xy2
x
ANSWER: –xy2 + x 8
How do you subtract polynomials? 1. Drop any parentheses around the 1st polynomial. They don’t matter. 2. Distribute the “” across the 2nd polynomial so you make sure the entire 2nd polynomial is subtracted, not just its first term. Clean up any stray “”s. 3. Combine like terms. Remember to take along any symbol on the left. 4. If necessary, clean things up for a fully simplified, standard form answer.
EXAMPLES: (6x2 + 8x) (3x2 + 5x) 6x2 + 8x
– 3x2 – 5x
ANSWER: 3x2 + 3x
(3xy2 + 4x 8) (xy2 7x) 3xy2 + 4x 8
xy2 + 7x
ANSWER: 2xy2 + 11x 8
© D. Stark 2017
10/3/2017
OVERVIEW: polynomials
2
How do you multiply monomials? 1. Multiply the coefficients and/or constants. 3(5x) = 15x
4(x) = 4(1x) = 4x
• If there’s no coefficient, it’s an invisible 1: x = 1x
x2 = 1x2
ab2 = 1ab2
2. Multiply each type of variable, either by expanding the exponents and counting or by following the product rule of exponents: When you’re multiplying powers with the same base, just add the exponents and keep the base. (x4)(x2) = (x)(x)(x)(x) • (x)(x) = x4 + 2 = x6 (x3)(x) = (x3)(x1) = x3 + 1 = x4 (x2yz3)(x5y4) = x7y5z3 [Note that in x2yz3 only the z is cubed, not the y.] • If there’s no exponent on a variable, it’s an invisible 1: x = x1 ab = a1b1 • If one monomial is missing a variable, just pretend it’s a 1. Multiplying by 1 doesn’t change anything. EXAMPLES: 2x3(4x2) = 8x5 –3x2y(–5x4y4) = 15x6y5 –4a3 • 6ab4 = –24a4b4
[ • is an alternative to ( ) meaning multiply.]
How do you multiply a monomial by a polynomial? Use the distributive property. EXAMPLES: 3x2(6x3 + 4) = 18x5 + 12x2 –7x2y(4xy – 3x) = –28x3y2 + 21x3y 4x(3x3y + 2x3 + 5) = 12x4y + 8x4 + 20x [The distributive property works with any number of terms inside the parentheses.]
© D. Stark 2017
10/3/2017
OVERVIEW: polynomials
3
How do you multiply binomials? [REALLY IMPORTANT!] Really, you just apply the distributive property twice—distributing one term of the 1st binomial and then the other. However, to help you make sure you haven’t forgotten any of the pieces, there are 2 popular methods: FOIL (the most common) and the box method.
FOIL method o Like PEMDAS, FOIL is an acronym for remembering an order of steps. EXAMPLE:
6x2 – 10x + 12x – 20
First Outer Inner Last
ANSWER: 6x2 + 2x – 20 Notice you’ll always have to combine the middle like terms to get the final answer.
–
box method 1. Draw a box and then cut it in two both horizontally & vertically. 2. Label the 2 top pieces with the 2 terms of the 1st polynomial, and the 2 side pieces with the 2 terms of the 2nd polynomial. Note that the subtraction symbol now becomes a negative sign. (You can use the + sign if you want.) 3. Find the area of each of the 4 little boxes by multiplying length width. Write each area in its little box. 4. Write your answer by listing the upper left box, the sum of the diagonals shown, and then the lower right box. If there’s no + or – shown, use a +.
ANSWER: 6x2 + 2x – 20
© D. Stark 2017
10/3/2017
OVERVIEW: polynomials
4
How do you divide monomials? * 1. Simplify any coefficients the way you would when reducing fractions. 2. Simplify each type of variable, either by expanding the exponents and canceling as with fractions or by following the quotient rule of exponents: When you’re dividing powers with the same base, just subtract the exponents and keep the base. EXAMPLE 1:
4𝑥 5 𝑦 6 𝑧 2
= 6𝑥 4 𝑦 2 𝑧 2
or
4 • 𝑥•𝑥•𝑥•𝑥•𝑥 • 𝑦•𝑦•𝑦•𝑦•𝑦•𝑦 • 𝑧•𝑧 6• 𝑥•𝑥•𝑥•𝑥 • 𝑦•𝑦 • 𝑧•𝑧
4 𝑥5 𝑦6 𝑧 2
2
6 𝑥4 𝑦2 𝑧
3
2=
𝑥𝑦 4
=
2 3
𝑥𝑦 4
x5 x4 = x5 – 4 = x1 = x y6 y2 = y6 – 2 = y4 z2 z2 = z0 = 1 [z 0]
EXAMPLE 2:
12𝑥 2 𝑦 5 3𝑥 4 𝑦 2 𝑧 2
=
4𝑦 3 𝑥2𝑧2
If there’s no exponent on a variable, it’s an invisible 1: x = x1 Any number divided by itself is 1:
12 12
=1
𝑥5 𝑥5
=
𝑥5 𝑥5
=1
Any non-zero number to the 0 power = 1: x5 x5= x5–5 = x0 = 1 if x 0 Whichever place (top or bottom) has a higher exponent for a variable is where the variable ends up (if there’s x6 on top and x4 on bottom, then x2 ends up on the top; if there’s x6 on bottom and x4 on top, then x2 ends up on the bottom). LEVEL 3 NOTE: If you end up with a negative exponent on top, drop the negative and put the power on the bottom. If you end up with a negative exponent on the bottom, drop the negative and put the power on the top. 𝑥 –5 𝑧 6 𝑧6 𝑧5 𝑥 4𝑧 5
𝑦2
=
𝑥 5 𝑦2
𝑥 –4 𝑦3
=
𝑦3
* If an expression has a variable in the denominator, it’s a rational expression, not a polynomial. A rational expression is a polynomial divided by a polynomial—like a giant algebraic fraction.
© D. Stark 2017
10/3/2017
OVERVIEW: polynomials
5
How do you divide a polynomial by a monomial? Splitting the fractions method 1. Split up each term of the numerator so it’s over the same denominator. 2. Reduce each term. EXAMPLE:
6𝑥 6 + 3𝑥 4 – 9𝑥 3
=
9𝑥 2 =
2 3
4
1
6𝑥 6 9𝑥 2
+
3𝑥 4 9𝑥 2
2
𝑥 + 𝑥 −𝑥
–
OR
3
9𝑥 3 9𝑥 2
=
2𝑥 4 3
6𝑥 6 9𝑥 2 𝑥2
+
3
3𝑥 4
+
9𝑥 2
–
9𝑥 3 9𝑥 2
−𝑥
Note that you can’t cancel factors from the top & bottom if there’s addition or subtraction going on. You can cancel only things being multiplied. Factor then cancel method (**) If you can factor either the numerator or the denominator, then sometimes you can cancel out a common factor on the top and bottom. EXAMPLE 1:
EXAMPLE 2:
𝑎–3 2𝑎 – 6
=
𝑎–3 2(𝑎 – 3)
3𝑐𝑑 2 + 3𝑐𝑑 𝑑+1
=
=
(𝑎 – 3) 2(𝑎 – 3)
3𝑐𝑑(𝑑 + 1) 𝑑+1
=
=
1 2
3𝑐𝑑(𝑑 + 1) 𝑑+1
= 3cd
How do you divide a polynomial by another polynomial? Believe it or not, you can do good old long division with polynomials! The other methods here should be enough for the GED test though. ** ADVANCED NOTE: The original rational expressions here have excluded values—values that would make the expression undefined since they would make the denominator 0. For graphing, you’d have to note those excluded values before canceling since they get lost in the final answer. This is beyond the scope of what we’re dealing with though. GED® is a registered trademark of the American Council on Education (ACE) and administered exclusively by GED Testing Service LLC under license. This material is not endorsed or approved by ACE or GED Testing Service.
© D. Stark 2017
10/3/2017
OVERVIEW: polynomials
6
What is a polynomial?
It’s a type of algebraic expression (one with no variables in the denominator or square roots).
It involves one or more terms combined by addition or subtraction. o If there’s subtraction, the term after it is considered negative.
The polynomial 3x2 + 4x 5 contains 3 terms: 3x2, 4x, and 5. This may be easier to see if you rewrite subtraction as the addition of a negative: 3x2 + 4x + 5
Polynomials are named by how many terms they have: Examples 2 2 monomial: 1 term 3y 𝑧 5a2b3 binomial: 2 terms trinomial: 3 terms
2x 4.5y 6x +
3
5y2 + 3z
2
𝑦 3z 5
3a2 + 2bc
4x2 + 3y2 xy
How do you evaluate polynomials?
2a2b + 3ab2 5ab
“Plug ’n play.”
(To evaluate an expression means to find its value, given the values of the variables.)
1. First plug in all the values on one line to replace the variables. o Use parentheses around the values you substitute. This will indicate multiplication and mark negative numbers. 2. Then crank out the rest with the order of operations, doing one step per line. EXAMPLES: Evaluate 6x – 3y + xy for x = 4 and y = 2
Evaluate 3x2 – 2y2 + xy for x = 5 and y = –3
6x – 3y + xy = 6(4) – 3(2) + (4)(2) = 24 – 6 + 8 = 26
3x2 – 2y2 + xy = 3(5)2 – 2(–3)2 + (5)(–3) = 3(25) – 2(9) + (5)(–3) = 75 – 18 + (–15) = 57 + (–15) = 42
© D. Stark 2017
10/3/2017
OVERVIEW: polynomials
1
How do you add polynomials? 1. Drop any parentheses. They don’t matter for addition. 2. Combine like terms. Remember to “Take what’s left”—take along any symbol on the left of a term as part of that term. 3. If necessary, clean things up for a fully simplified, standard form answer.
EXAMPLES: (2x2 + 4x) + (3x2 7x) 2x2
+ 4x
+ 3x2
– 7x
ANSWER: 5x2 3x
(–4xy2 + 2x 8) + (3xy2 x) 8
–4xy2 + 2x
+ 3xy2
x
ANSWER: –xy2 + x 8
How do you subtract polynomials? 1. Drop any parentheses around the 1st polynomial. They don’t matter. 2. Distribute the “” across the 2nd polynomial so you make sure the entire 2nd polynomial is subtracted, not just its first term. Clean up any stray “”s. 3. Combine like terms. Remember to take along any symbol on the left. 4. If necessary, clean things up for a fully simplified, standard form answer.
EXAMPLES: (6x2 + 8x) (3x2 + 5x) 6x2 + 8x
– 3x2 – 5x
ANSWER: 3x2 + 3x
(3xy2 + 4x 8) (xy2 7x) 3xy2 + 4x 8
xy2 + 7x
ANSWER: 2xy2 + 11x 8
© D. Stark 2017
10/3/2017
OVERVIEW: polynomials
2
How do you multiply monomials? 1. Multiply the coefficients and/or constants. 3(5x) = 15x
4(x) = 4(1x) = 4x
• If there’s no coefficient, it’s an invisible 1: x = 1x
x2 = 1x2
ab2 = 1ab2
2. Multiply each type of variable, either by expanding the exponents and counting or by following the product rule of exponents: When you’re multiplying powers with the same base, just add the exponents and keep the base. (x4)(x2) = (x)(x)(x)(x) • (x)(x) = x4 + 2 = x6 (x3)(x) = (x3)(x1) = x3 + 1 = x4 (x2yz3)(x5y4) = x7y5z3 [Note that in x2yz3 only the z is cubed, not the y.] • If there’s no exponent on a variable, it’s an invisible 1: x = x1 ab = a1b1 • If one monomial is missing a variable, just pretend it’s a 1. Multiplying by 1 doesn’t change anything. EXAMPLES: 2x3(4x2) = 8x5 –3x2y(–5x4y4) = 15x6y5 –4a3 • 6ab4 = –24a4b4
[ • is an alternative to ( ) meaning multiply.]
How do you multiply a monomial by a polynomial? Use the distributive property. EXAMPLES: 3x2(6x3 + 4) = 18x5 + 12x2 –7x2y(4xy – 3x) = –28x3y2 + 21x3y 4x(3x3y + 2x3 + 5) = 12x4y + 8x4 + 20x [The distributive property works with any number of terms inside the parentheses.]
© D. Stark 2017
10/3/2017
OVERVIEW: polynomials
3
How do you multiply binomials? [REALLY IMPORTANT!] Really, you just apply the distributive property twice—distributing one term of the 1st binomial and then the other. However, to help you make sure you haven’t forgotten any of the pieces, there are 2 popular methods: FOIL (the most common) and the box method.
FOIL method o Like PEMDAS, FOIL is an acronym for remembering an order of steps. EXAMPLE:
6x2 – 10x + 12x – 20
First Outer Inner Last
ANSWER: 6x2 + 2x – 20 Notice you’ll always have to combine the middle like terms to get the final answer.
–
box method 1. Draw a box and then cut it in two both horizontally & vertically. 2. Label the 2 top pieces with the 2 terms of the 1st polynomial, and the 2 side pieces with the 2 terms of the 2nd polynomial. Note that the subtraction symbol now becomes a negative sign. (You can use the + sign if you want.) 3. Find the area of each of the 4 little boxes by multiplying length width. Write each area in its little box. 4. Write your answer by listing the upper left box, the sum of the diagonals shown, and then the lower right box. If there’s no + or – shown, use a +.
ANSWER: 6x2 + 2x – 20
© D. Stark 2017
10/3/2017
OVERVIEW: polynomials
4
How do you divide monomials? * 1. Simplify any coefficients the way you would when reducing fractions. 2. Simplify each type of variable, either by expanding the exponents and canceling as with fractions or by following the quotient rule of exponents: When you’re dividing powers with the same base, just subtract the exponents and keep the base. EXAMPLE 1:
4𝑥 5 𝑦 6 𝑧 2
= 6𝑥 4 𝑦 2 𝑧 2
or
4 • 𝑥•𝑥•𝑥•𝑥•𝑥 • 𝑦•𝑦•𝑦•𝑦•𝑦•𝑦 • 𝑧•𝑧 6• 𝑥•𝑥•𝑥•𝑥 • 𝑦•𝑦 • 𝑧•𝑧
4 𝑥5 𝑦6 𝑧 2
2
6 𝑥4 𝑦2 𝑧
3
2=
𝑥𝑦 4
=
2 3
𝑥𝑦 4
x5 x4 = x5 – 4 = x1 = x y6 y2 = y6 – 2 = y4 z2 z2 = z0 = 1 [z 0]
EXAMPLE 2:
12𝑥 2 𝑦 5 3𝑥 4 𝑦 2 𝑧 2
=
4𝑦 3 𝑥2𝑧2
If there’s no exponent on a variable, it’s an invisible 1: x = x1 Any number divided by itself is 1:
12 12
=1
𝑥5 𝑥5
=
𝑥5 𝑥5
=1
Any non-zero number to the 0 power = 1: x5 x5= x5–5 = x0 = 1 if x 0 Whichever place (top or bottom) has a higher exponent for a variable is where the variable ends up (if there’s x6 on top and x4 on bottom, then x2 ends up on the top; if there’s x6 on bottom and x4 on top, then x2 ends up on the bottom). LEVEL 3 NOTE: If you end up with a negative exponent on top, drop the negative and put the power on the bottom. If you end up with a negative exponent on the bottom, drop the negative and put the power on the top. 𝑥 –5 𝑧 6 𝑧6 𝑧5 𝑥 4𝑧 5
𝑦2
=
𝑥 5 𝑦2
𝑥 –4 𝑦3
=
𝑦3
* If an expression has a variable in the denominator, it’s a rational expression, not a polynomial. A rational expression is a polynomial divided by a polynomial—like a giant algebraic fraction.
© D. Stark 2017
10/3/2017
OVERVIEW: polynomials
5
How do you divide a polynomial by a monomial? Splitting the fractions method 1. Split up each term of the numerator so it’s over the same denominator. 2. Reduce each term. EXAMPLE:
6𝑥 6 + 3𝑥 4 – 9𝑥 3
=
9𝑥 2 =
2 3
4
1
6𝑥 6 9𝑥 2
+
3𝑥 4 9𝑥 2
2
𝑥 + 𝑥 −𝑥
–
OR
3
9𝑥 3 9𝑥 2
=
2𝑥 4 3
6𝑥 6 9𝑥 2 𝑥2
+
3
3𝑥 4
+
9𝑥 2
–
9𝑥 3 9𝑥 2
−𝑥
Note that you can’t cancel factors from the top & bottom if there’s addition or subtraction going on. You can cancel only things being multiplied. Factor then cancel method (**) If you can factor either the numerator or the denominator, then sometimes you can cancel out a common factor on the top and bottom. EXAMPLE 1:
EXAMPLE 2:
𝑎–3 2𝑎 – 6
=
𝑎–3 2(𝑎 – 3)
3𝑐𝑑 2 + 3𝑐𝑑 𝑑+1
=
=
(𝑎 – 3) 2(𝑎 – 3)
3𝑐𝑑(𝑑 + 1) 𝑑+1
=
=
1 2
3𝑐𝑑(𝑑 + 1) 𝑑+1
= 3cd
How do you divide a polynomial by another polynomial? Believe it or not, you can do good old long division with polynomials! The other methods here should be enough for the GED test though. ** ADVANCED NOTE: The original rational expressions here have excluded values—values that would make the expression undefined since they would make the denominator 0. For graphing, you’d have to note those excluded values before canceling since they get lost in the final answer. This is beyond the scope of what we’re dealing with though. GED® is a registered trademark of the American Council on Education (ACE) and administered exclusively by GED Testing Service LLC under license. This material is not endorsed or approved by ACE or GED Testing Service.
© D. Stark 2017
10/3/2017
OVERVIEW: polynomials
6
