SAT Mathematics : Simplifying Polynomials

Study concepts, example questions & explanations for SAT Mathematics

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Example Questions

Example Question #1 : Simplifying Polynomials

Choose the answer which best simplifies the following expression:

\displaystyle (10x^{2} + 5x - 5) + (3x^2 - 4x + 8)

Possible Answers:

\displaystyle 13x^2 + 5x - 3

\displaystyle 7x^2 + 5x - 3

\displaystyle 7x^2 + x + 3

\displaystyle 13x^2 + x + 3

Correct answer:

\displaystyle 13x^2 + x + 3

Explanation:

To solve this problem simply remove the parentheses and add the like terms:

\displaystyle (10x^{2} + 5x - 5) + (3x^2 - 4x + 8)

\displaystyle 10x^2 + 3x^2 +5x-4x-5+8

\displaystyle 13x^2 +x + 3

Example Question #2 : Simplifying Polynomials

Choose the answer which best simplifies the following expression:

\displaystyle (10x^{2} + 5x - 5) + (3x^2 - 4x + 8)

Possible Answers:

\displaystyle 13x^2 + x + 3

\displaystyle 13x^2 + 5x - 3

\displaystyle 7x^2 + 5x - 3

\displaystyle 7x^2 + x + 3

Correct answer:

\displaystyle 13x^2 + x + 3

Explanation:

To solve this problem simply remove the parentheses and add the like terms:

\displaystyle (10x^{2} + 5x - 5) + (3x^2 - 4x + 8)

\displaystyle 10x^2 + 3x^2 +5x-4x-5+8

\displaystyle 13x^2 +x + 3

Example Question #3 : Simplifying Polynomials

Choose the answer which best simplifies the following expression:

\displaystyle (4y^2 +2y + 6) + (y^2 +3y - 10)

Possible Answers:

\displaystyle 3y^2 - 3y + 4

\displaystyle 5y^2 +5y +4

\displaystyle 3y^2 + 3y -4

\displaystyle 5y^2 +5y-4

Correct answer:

\displaystyle 5y^2 +5y-4

Explanation:

To simplify, simply remove the parentheses and combine like terms:

\displaystyle (4y^2 +2y +6) + (y^2 +3y -10)

\displaystyle 4y^2 +y^2 +2y +3y +6 -10

\displaystyle 5y^2 +5y - 4

Example Question #4 : Simplifying Polynomials

Choose the answer that best simplifies the following expression:

\displaystyle (12p^2 + 3p -12) + (-3p^2 - p +10)

Possible Answers:

\displaystyle 9p^2 +2p -10

\displaystyle 9p^2 - p +2

\displaystyle 9p^2 - 2p + 2

\displaystyle 9p^2 +2p -2

Correct answer:

\displaystyle 9p^2 +2p -2

Explanation:

To simplify, remove parentheses and combine like terms:

\displaystyle (12p^2 +3p -12) + (-3p^2 - p +10)

\displaystyle 12p^2 -3p^2 +3p - p -12 +10

\displaystyle 9p^2 +2p-2

Example Question #5 : Simplifying Polynomials

Choose the answer that best simplifies the following expression:

\displaystyle (q^2 +4q + 8) - (2q^2 -2q - 8)

Possible Answers:

\displaystyle -q^2-6q

\displaystyle 3q^2 +2q

\displaystyle q^2-6q+16

\displaystyle -q^2 +6q +16

Correct answer:

\displaystyle -q^2 +6q +16

Explanation:

To simplify, remove parentheses and combine like terms, but make sure to distribute the negative across all terms in the second set of parentheses, changing the sign of each:

\displaystyle q^2 + 4q + 8 - 2q^2 +2q + 8

Then combine like terms:

\displaystyle -q^2 +6q +16

 

Example Question #6 : Simplifying Polynomials

Choose the answer that best simplifies the following expression:

\displaystyle (5r^2 +3r - 5) - (4r^2+5r - 8)

Possible Answers:

\displaystyle r^2 -2r - 3

\displaystyle r^2 -2r +3

\displaystyle r^2 +2r +3

\displaystyle r^2 +2r -3

Correct answer:

\displaystyle r^2 -2r +3

Explanation:

To simplify, remove parentheses and combine like terms, remembering the ever-important step of applying the negative sign to each term within the second set of parentheses:

\displaystyle 5r^2 +3r - 5 -4r^2 -5r +8

\displaystyle 5r^2 -4r^2 +3r-5r-5+8

\displaystyle r^2 -2r +3

Example Question #7 : Simplifying Polynomials

Choose the answer that best simplifies the following expression:

\displaystyle (m^2 +2m + 2) - (3m^2 -2m + 4)

Possible Answers:

\displaystyle -2m^2 +4m -2

\displaystyle 2m^2 -4m + 2

\displaystyle 2m^2 - 2m + 2

\displaystyle -2m^2 -4m -2

Correct answer:

\displaystyle -2m^2 +4m -2

Explanation:

To simplify, remove parentheses and combine like terms, remembering to distribute the negative (by changing each sign) for the terms in the second set of parentheses:

\displaystyle m^2 +2m +2 -3m^2 +2m -4

\displaystyle m^2 -3m^2 +2m +2m +2 -4

\displaystyle -2m^2 +4m -2

Example Question #8 : Simplifying Polynomials

Simplify the following expression:

\displaystyle (2q^{2}+4q+7)-(3q^{2}-5)

Possible Answers:

\displaystyle -q^2+4q+2

\displaystyle -q^{2}+4q+12

\displaystyle -3q^{2}+12

\displaystyle -5q^{2}+4q+2

Correct answer:

\displaystyle -q^{2}+4q+12

Explanation:

This is not a FOIL problem, as we are adding rather than multiplying the terms in parentheses.

Add like terms together:

\displaystyle 2q^{2}-3q^{2}=-1q^{2}=-q^2

\displaystyle 4q has no like terms.

\displaystyle 7-(-5)=12

Combine these terms into one expression to find the answer:

\displaystyle -q^{2}+4q+12

Example Question #8 : Simplifying Polynomials

Simplify the following expression:

\displaystyle (9x^3-4x^2+3x-1)-(3x^3-2x^2-2x+3)

Possible Answers:

\displaystyle 6x^3+2x^2+x-2

\displaystyle 6x^3+6x^2+5x-4

\displaystyle 6x^3-6x^2+x+2

\displaystyle 6x^3-2x^2+5x-4

Correct answer:

\displaystyle 6x^3-2x^2+5x-4

Explanation:

To simplify this expression you need to carefully remove the parentheses by distributing the negative sign across all terms in the second set of parentheses. To do this, just take the opposite of each of those signs. That gives you:

\displaystyle 9x^3-4x^2+3x-1-3x^3+2x^2+2x-3

Then you can combine like terms:

\displaystyle 6x^3-2x^2+5x-4

Example Question #10 : Simplifying Polynomials

Choose the answer that best simplifies the following expression:

\displaystyle (2n^ 2 - 4n + 16) - (3n^2 + 4n -12)

Possible Answers:

\displaystyle n^2 + 8n +28

\displaystyle -n^2 - 8n + 28

\displaystyle -n^2 + 8n +28

\displaystyle -n^2 +8n -28

Correct answer:

\displaystyle -n^2 - 8n + 28

Explanation:

To solve this expression, you need to remove the parentheses, being careful to account for the negative prior to the second set of parentheses. To distribute that negative, take the opposite of each sign for that set of values. Then, combine like terms:

\displaystyle (2n^ 2 - 4n + 16) - (3n^2 + 4n -12)

\displaystyle 2n^ 2 -4n + 16 -3n^2 -4n +12

\displaystyle 2n^2 -3n^2 -4n -4n +16 +12

\displaystyle -n^2 -8n +28

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