Q5.67 - Intermediate Algebra

Question

Choose the appropriate pattern and use it to find product:

$$\begin{gathered} \left(a\right) (9b-2)(2b+9) \\ \left(b\right) {(9p-4)}^2 \\ \left(c\right) {(7y+1)}^2 \\ \left(d\right) (4r-3)(4r+3) \end{gathered}$$ 

Step-by-Step Solution

Answer

$$\begin{gathered} Multiply the binomials, we get \\ \left(a\right) 9b\times 2b+9b\times 9-2\times 2b-2\times 9 \\ 18b^2+77b-18 \\ \left(b\right) {\left(9p\right)}^2+{\left(4\right)}^2-2\times 9p\times 4 \\ 81p^2+16-72p \\ \left(c\right) {\left(7y\right)}^2+1+2\times 7y\times 1 \\ 49y^2+1+14y \\ \left(d\right) 4r\times 4r+4r\times 3-3\times 4r-3\times 3 \end{gathered}$$

Step 1

$$\begin{gathered} Given Information \\ \left(a\right) (9b-2)(2b+9) \\ \left(b\right) {(9p-4)}^2\left(c\right) {(7y+1)}^2\left(d\right) (4r-3)(4r+3) \end{gathered}$$

Step 2

$$\begin{gathered} A binomial conjugate is found simply by changing the sign of the 2nd term. T \\ he binomial conjugate of (a+b) is (a-b). \\ Vertical addition is a method of adding where the numbers are lined up in \\ The mnemonic F O I L tells us exactly what terms to multiply, and in what order: \\ First – multiply the first terms \\ Outside – multiply the outside/outer terms \\ Inside – multiply the inside/inner terms \\ Last – multiply the last termscolumns according to their places value. \end{gathered}$$

Part(a) Step1. Use FOIL method.

$$\begin{gathered} \left(a\right) (9b-2)(2b+9) \\ Use FOIL 9b\times 2b+9b\times 9-2\times 2b-2\times 9 \\ Simplify 18b^2+77b-18 \end{gathered}$$

Part(b) Step1. We are asked to square a binomial. It fits the binomial square pattern.

$$\begin{gathered} \left(b\right) \\ Use pattern {\left(9p\right)}^2+{\left(4\right)}^2-2\times 9p\times 4 \\ Simplify 81p^2+16-72p \end{gathered}$$

Part(c) Step1. Again binomial square pattern.

$$\begin{gathered} \left(c\right) \\ Use pattern {\left(7y\right)}^2+{\left(1\right)}^2+2\times 7y\times 1 \\ Simplify 49y^2+1+14y \end{gathered}$$

Part(d) Step1. If fits the product of conjugate pattern.

$$\begin{gathered} \left(d\right) \\ Use pattern {\left(4r\right)}^2-{\left(3\right)}^2 \\ Simplify 16r^2-9 \end{gathered}$$