Question

1. create a polynomial function with a zero of 5 with a multiplicity of 4, a zero of -7, and a zero of 4 with a multiplicity of 3.

study this table:
end behavior of polynomial function graphs

	odd degree	even degree
end behavior	↘️ ↗️	↗️ ↘️	↗️ ↗️	↘️ ↘️
number of turning points	n - 1 at most; each turning point is a local (or global) max. or min.

use $g(x) = x^4 - 6x^2 - 27$ to answer 24 - 26.

1. find all zeros. (hint: factor)
2. how many x-intercepts does the function $g(x)$ have?
3. state the end behavior of the function $g(x)$.

Explanation:

Question 23

Step1: Recall polynomial zero form

If \( r \) is a zero of a polynomial with multiplicity \( m \), then \( (x - r)^m \) is a factor.
Given zeros: \( 5 \) (multiplicity 4), \( -7 \) (multiplicity 1), \( 4 \) (multiplicity 3).
So factors are \( (x - 5)^4 \), \( (x + 7)^1 \), \( (x - 4)^3 \).

Step2: Form the polynomial

A polynomial with these zeros is \( f(x) = a(x - 5)^4(x + 7)(x - 4)^3 \), where \( a \) is a non - zero real number (usually \( a = 1 \) if not specified).

Step1: Let \( u=x^{2} \)

Rewrite \( g(x)=x^{4}-6x^{2}-27 \) as \( g(x)=u^{2}-6u - 27 \) (by substituting \( u = x^{2} \)).

Step2: Factor the quadratic

Factor \( u^{2}-6u - 27 \). We need two numbers that multiply to \( - 27 \) and add to \( -6 \). The numbers are \( -9 \) and \( 3 \). So \( u^{2}-6u - 27=(u - 9)(u+3) \).

Step3: Substitute back \( u = x^{2} \)

We get \( g(x)=(x^{2}-9)(x^{2}+3) \). Then factor \( x^{2}-9 \) as a difference of squares: \( x^{2}-9=(x - 3)(x + 3) \). So \( g(x)=(x - 3)(x + 3)(x^{2}+3) \).

Step4: Find zeros

Set \( g(x)=0 \).

• For \( (x - 3)=0 \), \( x = 3 \).
• For \( (x + 3)=0 \), \( x=-3 \).
• For \( x^{2}+3 = 0 \), \( x^{2}=-3 \), so \( x=\pm\sqrt{3}i \) (complex zeros).

Step1: Recall x - intercepts are real zeros

From question 24, the real zeros of \( g(x) \) are \( x = 3 \) and \( x=-3 \). The zeros from \( x^{2}+3 = 0 \) are complex (not real), so they do not correspond to x - intercepts.

Step2: Count real zeros

There are 2 real zeros, so there are 2 x - intercepts.

Answer:

\( f(x)=(x - 5)^4(x + 7)(x - 4)^3 \) (or with a non - zero constant factor \( a \) multiplied)

Question 24