Sylvan Learning

Factor polynomials

Objectives

Student will be able to:

factor the difference of two squares.
factor perfect square trinomials.
factor polynomials by finding the greatest common factor.

Exercises

Complete the exercises. Show all work.

1. When Dante's factorization of the following problem was marked incorrect by his teacher, he was puzzled because he thought he had done it correctly. Explain where Dante made his error.

Possible explanation: The sum of two squares $\inline x^2+4$ is not factorable. 1. Correct Incorrect Not Assigned

2. Consider a circle whose area is given by the formula: $A= \pi (x^2-12x+36)\ \textup{mm}^2$ Factor the formula to find an expression for the radius of the circle. Hint: The expression in parentheses is $\inline \inline r^2.$

The radius of the circle is $\inline x-6\textup{ mm}$ . $\inline \begin{align*}r^{2}&=(x^2-12x+36)\\r^2&=(x-6)^{2}\\r&=x-6 \end{align}$ Check student's work. 2. Correct Incorrect Not Assigned

3. If the Earth was cut at the equator, the surface area of the cut could be expressed as approximately the area of a circle with this formula: $\pi (16x^2-296x+1,\!369)\ \textup{mi}^2$ Factor the polynomial in the parenthesis to find an expression for the radius of the Earth in miles. Hint: $\inline 1,369=37^2$

The radius is $\inline \inline 4x-37 \textup{ mi}.$ $\inline \begin{align*}r^{2}&=(16x^2-296x+1,\!369)\\r^2&=(4x-37)^{2}\\r&=4x-37 \end{align}$ Check student's work. 3. Correct Incorrect Not Assigned

4. Sopan's square painting has an area expressed by this polynomial: $(x^2-6x+9)\ \textup{in.}^{2}$ Factor the polynomial to find an expression for the length of a side of the painting. (Remember, the area of a square is the product of its dimensions.)

The length of a side of the painting is $\inline x-3$ inches. $\inline \begin{align*}s^{2}&=(x^2-6x+9)\\s^2&=(x-3)^{2}\\s&=x-3 \end{align}$ Check student's work. 4. Correct Incorrect Not Assigned

5. Hoa is building a cover for her son's rectangular sandbox. The area of the cover can be expressed by this binomial: Factor to determine the dimensions of the cover.

The dimensions of the cover are $\inline (x+4)\textup{ cm by }(x-4)\ \textup{cm}$ . $\inline \inline x^{2}-16=(x+4)(x-4)$ Check student's work. 5. Correct Incorrect Not Assigned