Factoring Expressions - ISEE Upper Level Mathematics Achievement

1. What is the factored form of $x^2 + 6x - 16$ over the integers?

   $(x+8)(x-2)$. Find two integers that multiply to -16 and add to 6, yielding factors $(x + 8)$ and $(x - 2)$.

2. What is the greatest common factor (GCF) of $12x^3y^2$ and $18x^2y^5$?

   $6x^2y^2$. The GCF is found by taking the lowest powers of common prime factors (2 and 3) and variables ($x^2$ and $y^2$) from both terms.

3. Identify the factored form of $4x^2 - 25$.

   $(2x-5)(2x+5)$. Recognize $4x^2 - 25$ as a difference of squares, $(2x)^2 - 5^2 = (2x - 5)(2x + 5)$.

4. Identify the factored form of $9x^2 - 12x + 4$ as a perfect square.

   $(3x-2)^2$. The trinomial $9x^2 - 12x + 4$ matches the perfect square form $(3x - 2)^2$.

5. Identify the factored form of $2x^2 - 18$ as completely as possible.

   $2(x-3)(x+3)$. First factor out the GCF of 2, then factor the remaining difference of squares $x^2 - 9$.

6. What is the factored form of $2x^2 + 7x + 3$?

   $(2x+1)(x+3)$. Factor by finding integers where the product of leading coefficients and constants gives the quadratic, matching $(2x + 1)(x + 3)$.

7. What is the factored form of $x^2 + 10x + 25$ as a perfect square trinomial?

   $(x+5)^2$. The trinomial $x^2 + 10x + 25$ is a perfect square since it matches $(x + 5)^2$.

8. What is the factored form of $x^2 - x - 12$ over the integers?

   $(x-4)(x+3)$. Find two integers that multiply to -12 and add to -1, yielding factors $(x - 4)$ and $(x + 3)$.

9. Identify the factored form of $x^2 + 2x - 35$ over the integers.

   $(x+7)(x-5)$. Find two integers that multiply to -35 and add to 2, yielding factors $(x + 7)$ and $(x - 5)$.

10. Identify the factored form of $8x^2 - 12x$ using the GCF.

    $4x(2x-3)$. Factor out the GCF of $4x$ from both terms, leaving $2x - 3$ inside the parentheses.

11. State the identity for factoring a sum of cubes.

    $a^3+b^3=(a+b)(a^2-ab+b^2)$. The sum of cubes identity factors $a^3 + b^3$ into $(a + b)(a^2 - ab + b^2)$.

12. Identify the factored form of $5x^2 - 20x$ using the GCF.

    $5x(x-4)$. Factor out the GCF of $5x$ from both terms, leaving $x - 4$ inside the parentheses.

13. State the identity for factoring a difference of cubes.

    $a^3-b^3=(a-b)(a^2+ab+b^2)$. The difference of cubes identity factors $a^3 - b^3$ into $(a - b)(a^2 + ab + b^2)$.

14. State the identity for factoring a difference of squares.

    $a^2-b^2=(a-b)(a+b)$. The difference of squares identity factors $a^2 - b^2$ into $(a - b)(a + b)$.

15. What is the factored form of $3x^2 - 11x - 4$?

    $(3x+1)(x-4)$. Factor by finding integers where the product of leading coefficients and constants gives the quadratic, matching $(3x + 1)(x - 4)$.

16. What is the factored form of $x^2 - 16x + 64$ as a perfect square trinomial?

    $(x-8)^2$. The trinomial $x^2 - 16x + 64$ is a perfect square since it matches $(x - 8)^2$.

17. State the identity for factoring a perfect square trinomial $a^2-2ab+b^2$.

    $a^2-2ab+b^2=(a-b)^2$. The perfect square trinomial identity factors $a^2 - 2ab + b^2$ as $(a - b)^2$.

18. What is the factored form of $x^2 - 9$ as a difference of squares?

    $(x-3)(x+3)$. Recognize $x^2 - 9$ as a difference of squares, where $x^2 - 3^2 = (x - 3)(x + 3)$.

19. Identify the factored form of $6x^2 - 54$ by factoring out the GCF first.

    $6(x-3)(x+3)$. First factor out the GCF of 6, then factor the remaining difference of squares $x^2 - 9$.

20. State the identity for factoring a perfect square trinomial $a^2+2ab+b^2$.

    $a^2+2ab+b^2=(a+b)^2$. The perfect square trinomial identity factors $a^2 + 2ab + b^2$ as $(a + b)^2$.

21. What is the factored form of $ax+ay+bx+by$ by factoring by grouping?

    $(a+b)(x+y)$. Group terms as $(ax + ay) + (bx + by) = a(x + y) + b(x + y)$, then factor out the common binomial.

22. What is the factored form of $x^2 + 7x + 12$ over the integers?

    $(x+3)(x+4)$. Find two integers that multiply to 12 and add to 7, yielding factors $(x + 3)$ and $(x + 4)$.

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