✏️ Sum or Difference of Two Cubes
[tex] {\Large{\overline{\underline{\sf{\hookrightarrow Answer:}}}}} [/tex]
- The correct answer is letter B. The factors of 8q³ + 27 are (2q + 3) and (4q² - 6q + 9).
Solution:
The pattern of factoring the sum or difference of two cubes is:
- [tex] \sf{ a^3 + b^3 = (a + b)(a^2 - ab + b^2) } [/tex], and
- [tex] \sf{ a^3 - b^3 = (a - b)(a^2 + ab + b^2) } [/tex]
First find the cube root of the two terms.
- [tex] \sf{ \sqrt [3] {8q^3} = 2q } [/tex]
- [tex] \sf{ \sqrt [3] {27} = 3 } [/tex]
The sign of [tex] \sf{ b } [/tex] in the linear factor is the same as the sign in the middle of the expression being factored. Thus [tex] \sf{ \sf b } [/tex] is positive, and the linear factor is [tex] \sf{ \sf 2q + 3 } [/tex].
Now follow the pattern for the quadratic factor.
- [tex] \sf{ a^2 - ab + b^2 } [/tex]
- [tex] \sf{ \rightarrow (2q)^2 - (2q)(3) + (3)^2 } [/tex]
- [tex] \sf{ \rightarrow 4q^2 - 6q + 9 } [/tex]
Note that the sign of [tex] \sf{ \sf ab } [/tex] in the quadratic factor is the opposite of the sign of [tex] \sf{ \sf b } [/tex] in the linear factor, and the sign of [tex] \sf{ \sf b^2 } [/tex] is always positive.
Thus, the factored form of [tex] \sf{ 8q^3 + 27 } [/tex] is [tex] {\underline{\green{\sf{(2q + 3)(4q^2 - 6q + 9)}}}} [/tex]. The correct answer is letter B.
[tex]{\: \:}[/tex]
[tex] {\huge{\overline{\sf{Hope\:It\:Helps}}}} [/tex]
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