5.1.H: Homework

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    From LibreTexts, under CC BY 4.0, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    Practice Makes Perfect

    Square a Binomial Using the Binomial Squares Pattern

    In the following exercises, square each binomial using the Binomial Squares Pattern. Check your work by the expand and FOIL method.

    Exercise 1

    \((q+r)^2\)

    Answer

    \(q^2+2qr+r^2\)

    Exercise 2

    \((x+\frac{2}{3})^2\)

    Answer

    \(x^2+\frac{4}{3}x+\frac{4}{9}\)

    Exercise 3

    \((2y−3z)^2\)

    Answer

    \(4y^2−12yz+9z^2\)

    Exercise 4

    \((8p^3−3)^2\)

    Answer

    \(64p^6−48p^3+9\)

    Multiply Conjugates Using the Product of Conjugates Pattern (Difference of Squares)

    In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

    Exercise 5

    \((c−5)(c+5)\)

    Answer

    \(c^2−25\)

    Exercise 6

    \((8j+4)(8j−4)\)

    Answer

    \(64j^2−16\)

    Exercise 7

    \((7w+10x)(7w−10x)\)

    Answer

    \(49w^2−100x^2\)

    Exercise 8

    \((6m^3−4n^5)(6m^3+4n^5)\)

    Answer

    \(36m^6−16n^{10}\)

    Multiply special products relates to cubes

    In the following exercises, multiply as indicated.

    Exercise 9

    \((3x−2y)^3\)

    Answer

    \(27x^3−54x^2y+36xy^2−8y^3\)

    Exercise 10

    \((a+b)(a^2−ab+b^2)\)

    Answer

    \(a^3+b^3\)

    Exercise 11

    \((r−t)(r^2+rt+t^2)\)

    Answer

    \(r^3−t^3\)

    Exercise 12

    \((x−3)(x^2+3x+9)\)

    Answer

    \(x^3−27\)

    Exercise 13

    \((2y+5)(4y^2−10y+25)\)

    Answer

    \(8y^3+125\)

    ​​​​​​Recognize and Use the Appropriate Special Product Pattern

    In the following exercises, perform the indicated operation.

    Exercise 14

    a. \((2r+12)^2\)

    b. \((x−y)^2−(x−y)^2\)

    c. \((x+3)(x−3)^2\)

    d. \((7a+b)(a−7b)\)

    e. \((k−6)^2\)

    Answer

    a. \(4r^2+48r+144\)

    b. \(0\)

    c. \(x^3−3x^2−9x+27\)

    d. \(7a^2−48ab−7b^2\)

    e. \(k^2−12k+36\)

    Applications

    Solve the following.

    Exercise 15

    Communications Engineering Signal distortion, \(D\), in a transmission system can be modeled as:

    \(D^2=s^2+(n−i)^2\)

    where:

    • \(s\) is the signal strength
    • \(n\) is noise level
    • \(i\) is interference level

    Expand the formula using a special product.

    Answer

    \(D^2=s^2+n^2−2ni+i^2\)

    Exercise 16

    Physics The elastic potential energy, \(PE\), of a stretched spring is given by the following formula

    \(PE=\frac{1}{2}kx^2\)

    where \(k\) is the spring constant and \(x\) is the displacement of the stretched spring, If the displacement of the spring at time \(t\) is given by the equation \(x=2t+1\), find an equation for the potential energy in terms of \(k\) and \(t\) and expand the formula using a special product.

    Answer

    \(PE=2kt^2+2kt+\frac{1}{2}k\)

    Writing Exercises

    The following questions focus on one of the most common errors in algebra. Take your time working through each problem, and use them as an opportunity to strengthen your understanding and avoid this mistake in the future.

    Exercise 17

    Marta did the following work on her homework paper:

    \[\begin{array}{c} {(3−y)^2}\\ {3^2−y^2}\\ {9−y^2}\\ \nonumber \end{array}\]

    Explain what is wrong with Marta’s work.

    Exercise 18

    Use the order of operations to show that \((3+5)^2\) is 64, and then use that numerical example to explain why \((a+b)^2 \ne a^2+b^2\)

    Answer

    Answers will vary.

    This page titled 5.1.H: Homework was last modified on Mon, 06 Jul 2026 03:20:43 GMT and is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Kiersten Purves.