Section 2.2
Trinomial Squares
Whenever you multiply a binomial by itself twice, the resulting trinomial is called a perfect trinomial square.
Below we will multiply the binomial $x+3$ by itself twice--- that is we will expand $(x+3)(x+3)$.
\begin{align}
(x+3)(x+3)&=x^2+3x+3x+9\\
&=x^2+6x+9
\end{align}
So we say that $x^2+6x+9$ is a perfect trinomial square.
The technique to factoring perfect trinomial squares is in the formula $$A^2+2AB+B^2=(A+B)^2\qquad\text{ and }\qquad A^2-2AB+B^2=(A-B)^2$$ where $A$ is the base of the first term and $B$ is the base of the last term.
When attempting to factor an expression with three terms, you need to determine these two things:
- If the base of the first and last term are perfect squares.
- Then determine if the middle term 2 times the product of the base of the first and last term.
Examples:
a. $x^2+12x+36=(x+6)^2$ |
The
base of the first term is $x$, the base of the last term is 6,
and the middle term is $2(x\cdot 6)=12x.$ |
b. $c^2+2c+1=(c+1)^2$ |
The
base of the first term is $c$, the base of the last term is $1$,
and the middle term is $2(c\cdot 1)=2c.$ |
c. Fill in the parentheses so that $x^2+8x+\left(\qquad\right)$ is a perfect square.
Solution: The middle term is $8x$ and also $2\cdot Bx$. Therefore, $B=4$ and we have \begin{align} x^2+8x+\left(\qquad\right)&=x^2+8x+4^2\\ &=x^2+8x+16 \end{align}
Example: Factoring perfect square trinomials: Factoring perfect square trinomials
Exercises:
Factor each of the following completely.
\begin{array}{l l l}
1. \;\;{{x}^{2}}+4x+4\;\;& 2. \;\;{{x}^{2}}-6x+9\;\;& 3. \;\;{{x}^{2}}-16x+64\;\;\\
4. \;\;{{x}^{2}}+12x+36\;\;& 5. \;\;{{x}^{2}}+2x+1\;\;& 6. \;\;{{x}^{2}}+26x+169\;\;\\
7. \;\;{{x}^{2}}+8x+16\;\;& 8. \;\;{{x}^{2}}-20x+100\;\;& 9. \;\;{{x}^{2}}-10x+25\;\;\\
10. \;\;{{x}^{2}}-x+\dfrac{1}{4}\;\;& 11. \;\;{{x}^{2}}+3x+\dfrac{9}{4}\;\;& 12. \;\;4{{x}^{2}}-4x+1\;\;\\
13. \;\;9{{x}^{2}}-12x+4\;\;& 14. \;\;{{x}^{4}}-12{{x}^{2}}+36\;\;& 15. \;\;{{x}^{4}}+8{{x}^{2}}+16\;\;
\end{array}
Fill in each of the parentheses so that each of the following expressions is a perfect square.
\begin{array}{l l l}
16. \;\;{{x}^{2}}-8x+\left( {\;\;\;\;} \right)\;\;& 17. \;\;{{x}^{2}}+18x+\left( {\;\;\;\;} \right)\;\;& 18. \;\;{{x}^{2}}-14x+\left( {\;\;\;\;} \right)\;\;\\
19. \;\;{{x}^{2}}-6x+\left( {\;\;\;\;} \right)\;\;& 20. \;\;{{x}^{2}}+24x+\left( {\;\;\;\;} \right)\;\;& 21. \;\;{{x}^{2}}+40x+\left( {\;\;\;\;} \right)\;\;\\
22. \;\;{{x}^{2}}-10x+\left( {\;\;\;\;} \right)\;\;& 23. \;\;{{x}^{2}}+12x+\left( {\;\;\;\;} \right)\;\;& 24. \;\;{{x}^{4}}+22{{x}^{2}}+\left( {\;\;\;\;} \right)\;\;\\
25. \;\;{{x}^{2}}+\left( {\;\;\;\;} \right)x+25\;\;& 26. \;\;{{x}^{2}}+\left( {\;\;\;\;} \right)x+49\;\;& 27. \;\;{{x}^{2}}+\left( {\;\;\;\;} \right)x+4\;\;\\
28. \;\;{{x}^{2}}+\left( {\;\;\;\;} \right)x+9\;\;& 29. \;\;4{{x}^{2}}-4x+\left( {\;\;\;\;} \right)\;\;& 30. \;\;9{{x}^{2}}-6x+\left( {\;\;\;\;} \right)\;\;\\
31. \;\;9{{x}^{2}}-12x+\left( {\;\;\;\;} \right)\;\; 32.& \;\;{{x}^{2}}+11x+\left( {\;\;\;\;} \right)\;\;& 33. \;\;{{x}^{2}}-3x+\left( {\;\;\;\;} \right)\;\;\\
34. \;\;{{x}^{2}}-x+\left( {\;\;\;\;} \right)\;\;& 35. \;\;{{x}^{2}}+5x+\left( {\;\;\;\;} \right)\;\;& 36. \;\;{{x}^{2}}+7x+\left( {\;\;\;\;} \right)\;\;
\end{array}