Question

alberta diploma question
the following statements were made about the polynomial function
$y = (x - 5)^2(x + 3)$
i the x-intercepts are $-3$ and $5$.
ii the function has a degree of $3$.
iii the domain is $-3 \leq x \leq 5, x \in r$.
iv the range is $y \in r$
the statements that are true are
\bigcirc i and ii only
\bigcirc i, ii, and iv
\bigcirc iii and iv only
\bigcirc i, iii, and iv

Explanation:

Step1: Analyze Statement I

To find x - intercepts, set \( y = 0 \). So, \( 0=(x - 5)^{2}(x + 3) \). Using the zero - product property, \( (x - 5)^{2}=0\) gives \( x = 5 \) (with multiplicity 2) and \( x+3 = 0\) gives \( x=-3 \). So the x - intercepts are \( - 3 \) and \( 5 \). Statement I is true.

Step2: Analyze Statement II

Expand the polynomial \( y=(x - 5)^{2}(x + 3)=(x^{2}-10x + 25)(x + 3)=x^{3}-10x^{2}+25x+3x^{2}-30x + 75=x^{3}-7x^{2}-5x + 75 \). The degree of a polynomial is the highest power of \( x \). Here, the highest power of \( x \) is 3. So the function has a degree of 3. Statement II is true.

Step3: Analyze Statement III

The domain of a polynomial function \( y = a_{n}x^{n}+a_{n - 1}x^{n - 1}+\cdots+a_{1}x + a_{0}\), where \( n\) is a non - negative integer and \( a_{n}
eq0\), is all real numbers (\( x\in R\)), because we can plug in any real number for \( x \) and get a real number output. The domain is not restricted to \( - 3\leq x\leq5\). Statement III is false.

Step4: Analyze Statement IV

For a cubic polynomial (degree 3) with a positive leading coefficient (the coefficient of \( x^{3}\) is 1, which is positive), as \( x
ightarrow\infty\), \( y
ightarrow\infty\) and as \( x
ightarrow-\infty\), \( y
ightarrow-\infty\). So the range of the cubic polynomial is all real numbers (\( y\in R\)). Statement IV is true.

Answer:

I, II, and IV