6.10: Exercise Supplement

Exercise Supplement

Finding the factors of a Monomial

For the following problems, the first quantity represents the product and the second quantity represents a factor. Find the other factor.

Exercise $\PageIndex{1}$

$32a^4b,2b$

Answer

$16a^4$

Exercise $\PageIndex{2}$

$35x^3y^2,7x^3$

Exercise $\PageIndex{3}$

$44a^2b^2c,11b^2$

Answer

$4a^2c$

Exercise $\PageIndex{4}$

$50m^3n^5p^4q,10m^3q$

Exercise $\PageIndex{5}$

$51(a+1)^2(b+3)^4,3(a+1)$

Answer

$17(a+1)(b+3)^4$

Exercise $\PageIndex{6}$

$−26(x+2y)^3(x−y)^2,−13(x−y)$

Exercise $\PageIndex{7}$

$−8x^5y^4(x+y)^4(x+3y)^3,−2x(x+y)(x+3y)$

Answer

$4x^4y^4(x+y)^3(x+3y)^2$

Exercise $\PageIndex{8}$

$−(6a−5b)^{10}(7a−b)^8(a+3b)^7,−(6a−5b)^7(7a−b)^7(a+3b)^7$

Exercise $\PageIndex{9}$

$12x^{n+6}y^{2n-5}, -3x^{n+1}y^{n+3}$

Answer

$−4x^5y^{n−8}$

Exercise $\PageIndex{10}$

$−400a^{3n+10}b^{n−6}c^{4n+7},20a^{2n+8}c^{2n−1}$

Exercise $\PageIndex{11}$

$16x−32,16$

Answer

$(x−2)$

Exercise $\PageIndex{12}$

$35a−45,513$

Exercise $\PageIndex{13}$

$24a^2−6a,6a$

Answer

$4a−1$

Exercise $\PageIndex{14}$

$88x^4−33x^3+44x^2+55x,11x$

Exercise $\PageIndex{15}$

$9y^3−27y^2+36y,−3y$

Answer

$−3y^2+9y−12$

Exercise $\PageIndex{16}$

$4m^6−16m^4+16m^2,4m$

Exercise $\PageIndex{17}$

$−5x^4y^3+10x^3y^2−15x^2y^2,−5x^2y^2$

Answer

$x^2y−2x+3$

Exercise $\PageIndex{18}$

$−21a^5b^6c^4(a+2)^3+35a^5bc^5(a+2)^4,−7a^4b(a+2)^2$

Exercise $\PageIndex{19}$

$−x−2y−c^2,−1$

Answer

$x+2y+c^2$

Exercise $\PageIndex{20}$

$a+3b,−1$

Factoring a Monomial from a Polynomial ([link]) - The Greatest Common Factor ([link])

For the following problems, factor the polynomials.

Exercise $\PageIndex{21}$

$8a+4$

Answer

$4(2a+1)$

Exercise $\PageIndex{22}$

$10x+10$

Exercise $\PageIndex{23}$

$3y^2+27y$

Answer

$3y(y+9)$

Exercise $\PageIndex{24}$

$6a^2b^2+18a^2$

Exercise $\PageIndex{25}$

$21(x+5)+9$

Answer

$3(7x+38)$

Exercise $\PageIndex{26}$

$14(2a+1)+35$

Exercise $\PageIndex{27}$

$ma^3−m$

Answer

$m(a^3−1)$

Exercise $\PageIndex{28}$

$15y^3−24y+24$

Exercise $\PageIndex{29}$

$r^2(r+1)^3−3r(r+1)^2+r+1$

Answer

$(r+1)[r^2(r+1)^2−3r(r+1)+1]$

Exercise $\PageIndex{30}$

$Pa+Pb+Pc$

Exercise $\PageIndex{31}$

$(10−3x)(2+x)+3(10−3x)(7+x)$

Answer

$(10−3x)(23+4x)$

Factoring by Grouping

For the following problems, use the grouping method to factor the polynomials. Some may not be factorable.

Exercise $\PageIndex{32}$

$4ax+x+4ay+y$

Exercise $\PageIndex{33}$

$xy+4x−3y−12$

Answer

$(x−3)(y+4)$

Exercise $\PageIndex{34}$

$2ab−8b−3ab−12a$

Exercise $\PageIndex{35}$

$a^2−7a+ab−7b$

Answer

$(a+b)(a−7)$

Exercise $\PageIndex{36}$

$m^2+5m+nm+5n$

Exercise $\PageIndex{37}$

$r^2+rs−r−s$

Answer

$(r−1)(r+s)$

Exercise $\PageIndex{38}$

$8a^2bc+20a^2bc+10a^3b^3c+25a^3b^3$

Exercise $\PageIndex{39}$

$a(a+6)−(a+6)+a(a−4)−(a−4)$

Answer

$2(a+1)(a−1)$

Exercise $\PageIndex{40}$

$a(2x+7)−4(2x+7)+a(x−10)−4(x−10)$

Factoring Two Special Products - Factoring Trinomials with Leading Coefficient Other Than 1

For the following problems, factor the polynomials, if possible.

Exercise $\PageIndex{41}$

$m^2−36$

Answer

$(m+6)(m−6)$

Exercise $\PageIndex{42}$

$r^2−81$

Exercise $\PageIndex{43}$

$a^2+8a+16$

Answer

$(a+4)^2$

Exercise $\PageIndex{44}$

$c^2+10c+25$

Exercise $\PageIndex{45}$

$m^2+m+1$

Answer

not factorable

Exercise $\PageIndex{46}$

$r^2−r−6$

Exercise $\PageIndex{47}$

$a^2+9a+20$

Answer

$(a+5)(a+4)$

Exercise $\PageIndex{48}$

$s^2+9s+18$

Exercise $\PageIndex{49}$

$x^2+14x+40$

Answer

$(x+10)(x+4)$

Exercise $\PageIndex{50}$

$a^2−12a+36$

Exercise $\PageIndex{51}$

$n^2−14n+49$

Answer

$(n−7)^2$

Exercise $\PageIndex{52}$

$a^2+6a+5$

Exercise $\PageIndex{53}$

$a^2−9a+20$

Answer

$(a−5)(a−4)$

Exercise $\PageIndex{54}$

$6x^2+5x+1$

Exercise $\PageIndex{55}$

$4a^2−9a−9$

Answer

$(4a+3)(a−3)$

Exercise $\PageIndex{56}$

$4x^2+7x+3$

Exercise $\PageIndex{57}$

$42a^2+5a−2$

Answer

$(6a−1)(7a+2)$

Exercise $\PageIndex{58}$

$30y^2+7y−15$

Exercise $\PageIndex{59}$

$56m^2+26m+6$

Answer

$2(28m^2+13m+3)$

Exercise $\PageIndex{60}$

$27r^2−33r−4$

Exercise $\PageIndex{61}$

$4x^2+4xy−3y^2$

Answer

$(2x+3y)(2x−y)$

Exercise $\PageIndex{62}$

$25a^2+25ab+6b^2$

Exercise $\PageIndex{63}$

$2x^2+6x−20$

Answer

$2(x−2)(x+5)$

Exercise $\PageIndex{64}$

$−2y^2+4y+48$

Exercise $\PageIndex{65}$

$x^3+3x^2−4x$

Answer

$x(x+4)(x−1)$

Exercise $\PageIndex{66}$

$3y^4−27y^3+24y^2$

Exercise $\PageIndex{67}$

$15a^2b^2−ab−2b$

Answer

$b(15a^2b−a−2)$

Exercise $\PageIndex{68}$

$4x^3−16x^2+16x$

Exercise $\PageIndex{69}$

$18a^2 - 6a + \dfrac{1}{2}$

Answer

$(6a-1)(3a-\dfrac{1}{2})$

Exercise $\PageIndex{70}$

$a^4+16a^2b+16b^2$

Exercise $\PageIndex{71}$

$4x^2−12xy+9y^2$

Answer

$(2x−3y)^2$

Exercise $\PageIndex{72}$

$49b^4−84b^2+36$

Exercise $\PageIndex{73}$

$r^6s^8+6r^3s^4p^2q^6+9p^4q^{12}$

Answer

$(r^3s^4+3p^2q^6)^2$

Exercise $\PageIndex{74}$

$a^4−2a^2b−15b^2$

Exercise $\PageIndex{75}$

$81a^8b^{12}c^{10}−25x^{20}y^{18}$

Answer

$(9a^4b^6c^5+5x^{10}y^9)(9a^4b^6c^5−5x^{10}y^9)$