Lesson 15
The Remainder Theorem
 Let’s learn about the Remainder Theorem.
Problem 1
For the polynomial function \(f(x)=x^32x^25x+6\), we have \(f(0)=6, f(2)=\text4, f(\text2)=0, f(3)=0,f(\text1)=8, f(1)=0\). Rewrite \(f(x)\) as a product of linear factors.
Problem 2
Select all the polynomials that have \((x4)\) as a factor.
\(x^313x12\)
\(x^3 + 8x^2 + 19x + 12\)
\(x^3+6x + 5x  12\)
\(x^3x^210x8\)
\(x^24\)
Problem 3
Write a polynomial function, \(p(x)\), with degree 3 that has \(p(7) = 0\).
Problem 4
Long division was used here to divide the polynomial function \(p(x)=x^3+7x^220x110\) by \((x5)\) and to divide it by \((x+5)\).
\(\displaystyle \require{enclose} \begin{array}{r} x^2+12x+40 \\ x5 \enclose{longdiv}{x^3+7x^220x110} \\ \underline{x^3+5x^2} \phantom{20x110} \\ 12x^220x \phantom{110}\\ \underline{12x^2+60x} \phantom{110} \\ 40x110 \\ \underline{40x+200} \\ 90 \end{array}\)
\(\displaystyle \require{enclose} \begin{array}{r} x^2+2x30 \\ x+5 \enclose{longdiv}{x^3+7x^220x110} \\ \underline{x^35x^2} \phantom{20x110} \\ 2x^220x \phantom{110}\\ \underline{2x^210x} \phantom{110} \\ \text30x110 \\ \underline{30x+150} \\ 40 \end{array}\)

What is \(p(\text5)\)?

What is \(p(5)\)?
Problem 5
Which polynomial function has zeros when \(x=5,\frac23,\text7\)?
\(f(x)=(x+5)(2x+3)(x7)\)
\(f(x)=(x+5)(3x+2)(x7)\)
\(f(x)=(x5)(2x3)(x+7)\)
\(f(x)=(x5)(3x2)(x+7)\)
Problem 6
The polynomial function \(q(x)=3x^4+8x^313x^222x+24\) has known factors \((x+3)\) and \((x+2)\). Rewrite \(q(x)\) as the product of linear factors.
Problem 7
We know these things about a polynomial function \(f(x)\): it has degree 3, the leading coefficient is negative, and it has zeros at \(x=\text5,\text1, 3\). Sketch a graph of \(f(x)\) given this information.