Review these concepts at Factoring Expressions Study Guide.
Factoring Expressions Practice Questions
Problems
Practice 1
Factor each expression.
 6x + 2
 5x^{2} + x
 15y – 20y^{4}
 –2x^{5} + 4x^{4}
 13h^{2} – 13h
 a^{4} – a^{3} + a^{2}
 6s + 15s^{4}– 21s^{8}
 14z^{6}– 35z^{9}–42z^{4}
 9p^{3} + 27p^{5} – 6p^{10}+ 12p^{12}
 –20q^{8}–8q^{20} –40q^{12}– 80q^{10}
Practice 2
Factor each expression.
 18g^{8}h^{2}– 45g^{2}h^{8}
 –36x^{7}y^{3} + 12y^{2}
 5a^{9}bc^{3} – 11a^{6}b^{5}c^{5}
 6j^{11}k^{3}l^{2} + 48j^{7}k^{10}l^{4}– 28j^{9}k^{6}l^{6}
 50r^{5}t^{4} + 35s^{5}t^{4} + 45r^{5}s^{5}t^{4}
Solutions
Practice 1
 The factors of 6x are x, 1, 2, 3, and 6, and the factors of 2 are 1 and 2. Both numbers have 2 as a common factor, so we can factor a 2 out of each term.
Only the first term has an x, so we cannot factor out an x.
Divide both terms by 2: 6x ÷ 2 = 3x, and 2 ÷ 2 = 1.
Write 2 outside the parentheses to show that both terms are multiplied by it: 2(3x + 1).
 The factors of 5x^{2} are x^{2}, 1, and 5, and the factors of x are 1 and x. Both terms have x as a common factor, so we can factor an x out of each term.
Divide both terms by 5x^{2} ÷ x = 5x, and x ÷ x = 1.
Write x outside the parentheses to show that both terms are multiplied by it: x(5x + 1).
 The factors of 15y are y, 1, 3, 5, and 15, and the factors of 20y4 are y4, 1, 2, 4, 5, 10, and 20. Both terms have 5 and y as common factors, so we can factor 5y out of each term.
Divide both terms by 5y: 15y ÷ 5y = 3, and 20y^{4} ÷ 5y = 4y^{3}.
Write 5y outside the parentheses to show that both terms are multiplied by it: 5y(3 – 4y^{3}).
 The factors of –2x^{5} are –1, 2, x^{5}, and the factors of 4x^{4} are 1, 2, 4, and x^{4}.
Both terms have 2 and x as common factors. In fact, we can factor 2x^{4} out of both terms.
Divide both terms by 2x^{4}: –2x^{5} ÷ 2x^{4} = –x, and 4x^{4} ÷ 2x^{4} = 2.
–2x^{5} + 4x^{4} factors into 2x^{4}(–x + 2).
 The factors of 13h^{2} are h^{2}, 1, and 13, and the factors of 13h are h, 1, and 13.
Both terms have 13 and h as common factors.
Divide both terms by 13h: 13h^{2} ÷ 13h = h, and 13h ÷ 13h = 1.
13h^{2} – 13h factors into 13h(h – 1).
 Each term in a^{4} – a^{3} + a^{2} has no coefficient, but a is a common factor of every term. The smallest exponent of a is 2, in the term a^{2}.
Divide every term by a^{2}: a^{4} ÷ a^{2} = a^{2}, a^{3} ÷ a^{2} = a, and a^{2} ÷ a^{2} = 1.
a^{4} – a^{3} + a^{2} factors into a^{2}(a^{2} – a + 1).
 The greatest common factor of 6, 15, and 21 is 3. Every term has s as a common factor, and the smallest exponent of s is 1, in the term 6s.
Divide every term by 3s: 6s ÷ 3s = 2, 15s^{4} ÷ 3s = 5s^{3}, and 21s^{8} ÷ 3s = 7s^{7}.
6s + 15s^{4} – 21s^{8} factors into 3s(2 + 5s^{3} – 7s^{7}).
 The greatest common factor of 14, 35, and 42 is 7. Every term has z as a common factor, and the smallest exponent of z is 4, in the term 42z^{4}.
Divide every term by 7z^{4}:14z^{6} ÷ 7z^{4} = 2z^{2}, 35z^{9} ÷ 7z^{4} = 5z^{5}, and 42z^{4} ÷ 7z^{4} = 6.
14z^{6} – 35z^{9} – 42z^{4} factors into 7z^{4}(2z^{2} – 5z^{5} – 6).
 The greatest common factor of 9, 27, 6, and 12 is 3. Every term has p as a common factor, and the smallest exponent of p is 3, in the term 9p^{3}.
Divide every term by 3p^{3}: 3p^{3} ÷ 3p^{3} = 3, 27p^{5} ÷ 3p^{3} = 9p^{3}, 6p^{10} ÷ 3p^{3} = 2p7,and 12p^{12} ÷ 3p^{3} = 4p^{9}.
9p^{3} + 27p^{5} – 6p^{10} + 12p^{12} factors into 3p^{3}(3 + 9p^{2} – 2p^{7} + 4p^{9}).
 The greatest common factor of 20, 8, 40, and 80 is 4. Since every term is negative, –1 should be factored out of the expression, too. Every term has p as a common factor, and the smallest exponent of q is 8, in the term –20q^{8}.
Divide every term by –4q^{8}: –20q^{8} ÷ –4q^{8} = 5, –8q^{20} ÷ –4q^{8} = 2q^{12}, –40q^{12} ÷ –4q^{8} = 10q^{4}, and –80q^{10} ÷ –4q^{8} = 20q^{10}.
–20q^{8} – 8q^{20} – 40q12 – 80q^{10} factors into –4q^{8}(5 + 2q^{12} + 10q^{4} + 20q^{2}).

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