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.
\(32a^4b,2b\)
- Answer
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\(16a^4\)
\(35x^3y^2,7x^3\)
\(44a^2b^2c,11b^2\)
- Answer
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\(4a^2c\)
\(50m^3n^5p^4q,10m^3q\)
\(51(a+1)^2(b+3)^4,3(a+1)\)
- Answer
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\(17(a+1)(b+3)^4\)
\(−26(x+2y)^3(x−y)^2,−13(x−y)\)
\(−8x^5y^4(x+y)^4(x+3y)^3,−2x(x+y)(x+3y)\)
- Answer
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\(4x^4y^4(x+y)^3(x+3y)^2\)
\(−(6a−5b)^{10}(7a−b)^8(a+3b)^7,−(6a−5b)^7(7a−b)^7(a+3b)^7\)
\(12x^{n+6}y^{2n-5}, -3x^{n+1}y^{n+3}\)
- Answer
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\(−4x^5y^{n−8}\)
\(−400a^{3n+10}b^{n−6}c^{4n+7},20a^{2n+8}c^{2n−1}\)
\(16x−32,16\)
- Answer
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\((x−2)\)
\(35a−45,513\)
\(24a^2−6a,6a\)
- Answer
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\(4a−1\)
\(88x^4−33x^3+44x^2+55x,11x\)
\(9y^3−27y^2+36y,−3y\)
- Answer
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\(−3y^2+9y−12\)
\(4m^6−16m^4+16m^2,4m\)
\(−5x^4y^3+10x^3y^2−15x^2y^2,−5x^2y^2\)
- Answer
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\(x^2y−2x+3\)
\(−21a^5b^6c^4(a+2)^3+35a^5bc^5(a+2)^4,−7a^4b(a+2)^2\)
\(−x−2y−c^2,−1\)
- Answer
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\(x+2y+c^2\)
\(a+3b,−1\)
Factoring a Monomial from a Polynomial ( [link] ) - The Greatest Common Factor ( [link] )
For the following problems, factor the polynomials.
\(8a+4\)
- Answer
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\(4(2a+1)\)
\(10x+10\)
\(3y^2+27y\)
- Answer
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\(3y(y+9)\)
\(6a^2b^2+18a^2\)
\(21(x+5)+9\)
- Answer
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\(3(7x+38)\)
\(14(2a+1)+35\)
\(ma^3−m\)
- Answer
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\(m(a^3−1)\)
\(15y^3−24y+24\)
\(r^2(r+1)^3−3r(r+1)^2+r+1\)
- Answer
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\((r+1)[r^2(r+1)^2−3r(r+1)+1]\)
\(Pa+Pb+Pc\)
\((10−3x)(2+x)+3(10−3x)(7+x)\)
- Answer
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\((10−3x)(23+4x)\)
Factoring by Grouping
For the following problems, use the grouping method to factor the polynomials. Some may not be factorable.
\(4ax+x+4ay+y\)
\(xy+4x−3y−12\)
- Answer
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\((x−3)(y+4)\)
\(2ab−8b−3ab−12a\)
\(a^2−7a+ab−7b\)
- Answer
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\((a+b)(a−7)\)
\(m^2+5m+nm+5n\)
\(r^2+rs−r−s\)
- Answer
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\((r−1)(r+s)\)
\(8a^2bc+20a^2bc+10a^3b^3c+25a^3b^3\)
\(a(a+6)−(a+6)+a(a−4)−(a−4)\)
- Answer
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\(2(a+1)(a−1)\)
\(a(2x+7)−4(2x+7)+a(x−10)−4(x−10)\)
Factoring Two Special Products - Factoring Trinomials with Leading Coefficient Other Than 1
For the following problems, factor the polynomials, if possible.
\(m^2−36\)
- Answer
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\((m+6)(m−6)\)
\(r^2−81\)
\(a^2+8a+16\)
- Answer
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\((a+4)^2\)
\(c^2+10c+25\)
\(m^2+m+1\)
- Answer
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not factorable
\(r^2−r−6\)
\(a^2+9a+20\)
- Answer
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\((a+5)(a+4)\)
\(s^2+9s+18\)
\(x^2+14x+40\)
- Answer
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\((x+10)(x+4)\)
\(a^2−12a+36\)
\(n^2−14n+49\)
- Answer
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\((n−7)^2\)
\(a^2+6a+5\)
\(a^2−9a+20\)
- Answer
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\((a−5)(a−4)\)
\(6x^2+5x+1\)
\(4a^2−9a−9\)
- Answer
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\((4a+3)(a−3)\)
\(4x^2+7x+3\)
\(42a^2+5a−2\)
- Answer
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\((6a−1)(7a+2)\)
\(30y^2+7y−15\)
\(56m^2+26m+6\)
- Answer
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\(2(28m^2+13m+3)\)
\(27r^2−33r−4\)
\(4x^2+4xy−3y^2\)
- Answer
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\((2x+3y)(2x−y)\)
\(25a^2+25ab+6b^2\)
\(2x^2+6x−20\)
- Answer
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\(2(x−2)(x+5)\)
\(−2y^2+4y+48\)
\(x^3+3x^2−4x\)
- Answer
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\(x(x+4)(x−1)\)
\(3y^4−27y^3+24y^2\)
\(15a^2b^2−ab−2b\)
- Answer
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\(b(15a^2b−a−2)\)
\(4x^3−16x^2+16x\)
\(18a^2 - 6a + \dfrac{1}{2}\)
- Answer
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\((6a-1)(3a-\dfrac{1}{2})\)
\(a^4+16a^2b+16b^2\)
\(4x^2−12xy+9y^2\)
- Answer
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\((2x−3y)^2\)
\(49b^4−84b^2+36\)
\(r^6s^8+6r^3s^4p^2q^6+9p^4q^{12}\)
- Answer
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\((r^3s^4+3p^2q^6)^2\)
\(a^4−2a^2b−15b^2\)
\(81a^8b^{12}c^{10}−25x^{20}y^{18}\)
- Answer
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\((9a^4b^6c^5+5x^{10}y^9)(9a^4b^6c^5−5x^{10}y^9)\)