Problems 1-4 : Determine whether each trinomial is a perfect square. If so, factor. If not, explain.
Problem 1 :
x^{2} + 12x + 36
Problem 2 :
9x^{2} + 12x + 4
Problem 3 :
4x^{2} - 12x + 9
Problem 4 :
x^{2} + 9x + 16
Problems 5-8 : Determine whether each binomial is a difference of two squares. If so, factor. If not, explain.
Problem 5 :
x^{2} - 81
Problem 6 :
4x^{2} - 9
Problem 7 :
9p^{4} - 16q^{2}
Problem 8 :
x^{6} - 7y^{2}
Problems 9-12 : Factor the given trinomial.
Problem 9 :
x^{2} + 6x + 9
Problem 10 :
4y^{2} + 20y + 25
Problem 11 :
81z^{2} - 18z + 1
Problem 12 :
100k^{2} - 140k + 49
Problems 13-16 : Factor the given binomial.
Problem 13 :
x^{2} - 144
Problem 14 :
25y^{2} - 169
Problem 15 :
9a^{2} - 16b^{2}
Problem 16 :
c^{4} - d^{4}
1. Answer :
x^{2} + 12x + 36
The trinomial is a perfect square. Factor.
x^{2} + 12x + 36
a = x, b = 6.
Write the trinomial as a^{2} + 2ab + b^{2}.
= x^{2} + 2(x)(6) + 6^{2}
Write the trinomial as (a + b)^{2}.
= (x + 6)^{2}
2. Answer :
9x^{2} + 12x + 4
The trinomial is a perfect square. Factor.
9x^{2} + 12x + 4
a = 3x, b = 2.
Write the trinomial as a^{2} + 2ab + b^{2}.
= (3x)^{2} + 2(3x)(2) + 2^{2}
Write the trinomial as (a + b)^{2}.
= (3x + 2)^{2}
3. Answer :
4x^{2} - 12x + 9
The trinomial is a perfect square. Factor.
= 4x^{2} - 12x + 9
a = 2x, b = 3.
Write the trinomial as a^{2} - 2ab + b^{2}.
= (2x)^{2} - 2(2x)(3) + 3^{2}
Write the trinomial as (a - b)^{2}.
= (2x - 3)^{2}
4. Answer :
x^{2} + 9x + 16
2(x · 4) ≠ 9x
x^{2} + 9x + 16 is not a perfect-square trinomial because
9x ≠ 2(x · 4)
5. Answer :
x^{2} - 81
The polynomial is a difference of two squares.
= x^{2} - 81
= x^{2} - 9^{2}
a = x and b = 9, write the polynomial as (a + b)(a - b).
= (x + 9)(x - 9)
6. Answer :
4x^{2} - 9
The polynomial is a difference of two squares.
= 4x^{2} - 9
= (2x)^{2} - 3^{2}
a = 2x and b = 3, write the polynomial as (a + b)(a - b).
= (2x + 3)(2x - 3)
7. Answer :
9p^{4} - 16q^{2}
The polynomial is a difference of two squares.
= 9p^{4} - 16q^{2}
= (3p^{2})^{2} - (4q)^{2}
a = 3p^{2} and b = 4q, write the polynomial as (a + b)(a - b).
= (3p^{2} + 4q)(3p^{2} - 4q)
8. Answer :
x^{6} - 7y^{2}
7y^{2} is not a perfect square.
x^{6} - 7y^{2 }is not the difference of two squares because 7y^{2 }is not a perfect square.
9. Answer :
= x^{2} + 6x + 9
= x^{2} + 2(x)(3) + 3^{2}
Since the above expression is in the form of a^{2} + 2ab + b^{2}, it can be written in the form (a + b)^{2}.
= (x + 3)^{2}
10. Answer :
= 4y^{2} + 20y + 25
= 2^{2}y^{2} + 20y + 5^{2}
= (2y)^{2} + 2(2y)(5) + 5^{2}
Since the above expression is in the form of a^{2} + 2ab + b^{2}, it can be written in the form (a + b)^{2}.
= (2y + 5)^{2}
11. Answer :
81z^{2} - 18z + 1
= 9^{2}z^{2} - 18z + 1^{2}
= 9^{2}z^{2} - 2(9z)(1) + 1^{2}
= (9z)^{2} - 2(9z)(1) + 1^{2}
Since the above expression is in the form of a^{2} - 2ab + b^{2}, it can be written in the form (a - b)^{2}.
= (9z - 1)^{2}
12. Answer :
100k^{2} - 140k + 49
= 10^{2}k^{2} - 140k + 7^{2}
= (10k)^{2} - 2(10k)(7) + 7^{2}
Since the above expression is in the form of a^{2} - 2ab + b^{2}, it can be written in the form (a - b)^{2}.
= (10k - 7)^{2}
13. Answer :
= x^{2} - 144
= x^{2} - 12^{2}
The above binomial is a difference of two squares and it is in the form of (a^{2} - b^{2}). Take a = x and b = 12 and write the above binomial in the factored form (a + b)(a - b).
= (x + 12)(x - 12)
14. Answer :
= 25y^{2} - 169
= 5^{2}y^{2} - 13^{2}
= (5y)^{2} - 13^{2}
The above binomial is a difference of two squares and it is in the form of (a^{2} - b^{2}). Take a = 5y and b = 13 and write the above binomial in the factored form (a + b)(a - b).
= (5y + 13)(5y - 13)
15. Answer :
= 9a^{2} - 16b^{2}
= 3^{2}a^{2} - 4^{2}b^{2}
= (3a)^{2} - (4b)^{2}
The above binomial is a difference of two squares and it is in the form of (a^{2} - b^{2}). Take a = 3a and b = 4b and write the above binomial in the factored form (a + b)(a - b).
= (3a + 4b)(3a - 4b)
16. Answer :
= c^{4} - d^{4}
= (c^{2})^{2} - (d^{2})^{2}
= (c^{2})^{2} - (d^{2})^{2}
The above binomial is a difference of two squares and it is in the form of (a^{2} - b^{2}). Take a = c^{2} and b = d^{2} and write the above binomial in the factored form (a + b)(a - b).
= (c^{2} + d^{2})(c^{2} - d^{2})
Since (c^{2} - d^{2}) is a difference of two squares, it can be factored as (c + d)(c - d).
= (c^{2} + d^{2})(c + d)(c - d)
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