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Mathematics LibreTexts

1.3.1: Exercises 1.3

  • Page ID
    62180
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    Terms and Concepts

    Exercise \(\PageIndex{1}\)

    In your own words, explain the relationship between factors and roots.

    Answer

    Answers will vary.

    Exercise \(\PageIndex{2}\)

    If \(x=2\), \(x=5\),and \(x=-1\) are the only roots of the function \(f(x)\), what are the factors of \(f(x)\)?

    Answer

    The factors are \(x-2\), \(x-5\), and \(x+1\).

    Exercise \(\PageIndex{3}\)

    What does it mean for a quadratic to be irreducible?

    Answer

    It means it has no real number roots and that it cannot be factored.

    Exercise \(\PageIndex{4}\)

    What is the maximum number of linear factors that \(g(t) = t^6+t^4-2t^2 +1\) could have?

    Answer

    It can have a maximum of 6 linear factors since it is a sixth degree polynomial.

    Exercise \(\PageIndex{5}\)

    What is the maximum number of roots that \(g(t) = t^6+t^4-2t^2 +1\) could have?

    Answer

    It can have a maximum of 6 roots since it is a sixth degree polynomial.

    Problems

    In exercises \(\PageIndex{6}\) - \(\PageIndex{12}\), expand and simplify the given expressions.

    Exercise \(\PageIndex{6}\)

    \(3a(2b+5)(a-2b)\)

    Answer

    \(6a^2b-12ab^2+15a^2-30ab\)

    Exercise \(\PageIndex{7}\)

    \((2t+7)^2\)

    Answer

    \(4t^2 + 28t+49\)

    Exercise \(\PageIndex{8}\)

    \(2(x^2+3x+4)(2x+3)\)

    Answer

    \(4x^3+18x^2+34x+24\)

    Exercise \(\PageIndex{9}\)

    \((t^2-3t+1)(2t)-(t^2+2)(2t-3)\)

    Answer

    \(-3t^2-2t+6\)

    Exercise \(\PageIndex{10}\)

    \((x^3+x-2)(2)-(2x+1)(3x^2+1)\)

    Answer

    \(-4x^3-3x^2-5\)

    Exercise \(\PageIndex{11}\)

    \((x^2+3x-1)(4x)+(2x^2-5)(2x+3)\)

    Answer

    \(8x^3+18x^2-14x-15\)

    Exercise \(\PageIndex{12}\)

    \(3(\theta^2+4)^2(2\theta)\)

    Answer

    \(6\theta^5+48\theta^3+96\theta\)

    In exercises \(\PageIndex{13}\) - \(\PageIndex{20}\), factor each function completely.

    Exercise \(\PageIndex{13}\)

    \(g(x)=4x^2+4x+1\)

    Answer

    \(g(x)=(2x+1)^2\)

    Exercise \(\PageIndex{14}\)

    \(y(z)=z^2-7z+10\)

    Answer

    \(y(z)=(z-5)(z-2)\)

    Exercise \(\PageIndex{15}\)

    \(f(k)=k^4-27k\)

    Answer

    \(f(k)=k(k-3)(k^2+3k+9)\)

    Exercise \(\PageIndex{16}\)

    \(\theta(\gamma) = 6\gamma^2-\gamma -2\)

    Answer

    \(\theta(\gamma) = (3\gamma-2)(2\gamma+1)\)

    Exercise \(\PageIndex{17}\)

    \(x(z) = 3z^3+6z^2-24z\)

    Answer

    \(x(z)=3z(z+4)(z-2)\)

    Exercise \(\PageIndex{18}\)

    \(y(x)=x^3+8\)

    Answer

    \(y(x)=(x+2)(x^2-2x+4)\)

    Exercise \(\PageIndex{19}\)

    \(f(x)=2x^3 -x^2 -5x-2\)

    Answer

    \(f(x)=(2x+1)(x-2)(x+1)\)

    Exercise \(\PageIndex{20}\)

    \(f(y)=y^3-5y^2-2y+24\)

    Answer

    \(f(y)=(y-3)(y-4)(y+2)\)

    In exercises \(\PageIndex{21}\) - \(\PageIndex{24}\), determine the difference quotient of the given function.

    Exercise \(\PageIndex{21}\)

    \(g(t)=t^3+1\)

    Answer

    \(3t^2+3th+h^2\)

    Exercise \(\PageIndex{22}\)

    \(y(x)=2x^2-5\)

    Answer

    \(4x+2h\)

    Exercise \(\PageIndex{23}\)

    \(f(x)=x^3+x^2 -x\)

    Answer

    \(3x^2+3xh+h^2+2x+h-1\)

    Exercise \(\PageIndex{24}\)

    \(g(x)=4x^2+2x\)

    Answer

    \(8x+4h^2+2\)

    In exercises \(\PageIndex{25}\) - \(\PageIndex{27}\), find all real roots of the given function.

    Exercise \(\PageIndex{25}\)

    \(g(x)=4x^2+2x\)

    Answer

    \(x=0\) and \(x=-\frac{1}{2}\)

    Exercise \(\PageIndex{26}\)

    \(f(x)=x^3+x^2 -x\)

    Answer

    \(x=0\), \(x=\frac{-1+\sqrt{5}}{2}\), and \(x=\frac{-1-\sqrt{5}}{2}\)

    Exercise \(\PageIndex{27}\)

    \(y(x)=4x^2-5\)

    Answer

    \(x=\frac{\sqrt{5}}{2}\) and \(x=-\frac{\sqrt{5}}{2}\)

    In exercises \(\PageIndex{28}\) - \(\PageIndex{30}\), factor the given function, and relate the factors with the roots found in exercises \(\PageIndex{25}\) - \(\PageIndex{27}\).

    Exercise \(\PageIndex{28}\)

    \(g(x)=4x^2+2x\)

    Answer

    \(g(x) = 2x(2x+1)\); the factor \(x\) pairs with the root \(x=0\) and the factor \(2x+1\) pairs with the root \(x=-\frac{1}{2}\)

    Exercise \(\PageIndex{29}\)

    \(f(x)=x^3+x^2 -x\)

    Answer

    \(f(x) = x(x+ \frac{1-\sqrt{5}}{2})(x+\frac{1+\sqrt{5}}{2})\); the factor \(x\) pairs with the root \(x=0\), the factor \(x + \frac{1-\sqrt{5}}{2}\) pairs with the root \(x=\frac{-1+\sqrt{5}}{2}\), and the factor \(x+\frac{1-\sqrt{5}}{2}\) pairs with the root \(x=\frac{-1-\sqrt{5}}{2}\)

    Exercise \(\PageIndex{30}\)

    \(y(x)=4x^2-5\)

    Answer

    \(y(x) = 4(x-\frac{\sqrt{5}}{2})(x+\frac{\sqrt{5}}{2})\); the factor \(x-\frac{\sqrt{5}}{2}\) pairs with the root \(x=\frac{\sqrt{5}}{2}\) and the factor \(x+\frac{\sqrt{5}}{2}\) pairs with the root \(x=-\frac{\sqrt{5}}{2}\)


    This page titled 1.3.1: Exercises 1.3 is shared under a CC BY-NC license and was authored, remixed, and/or curated by Amy Givler Chapman, Meagan Herald, Jessica Libertini.

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