Question

the polynomial of degree 4, ( p(x) ), has a root of multiplicity 2 at ( x = 2 ) and roots of multiplicity 1 at ( x = 0 ) and ( x = -2 ). it goes through the point ( (5, 94.5) ). find a formula for ( p(x) ). ( p(x) = ) blank question help: video written example submit question

Explanation:

Step1: Write the general form of the polynomial

Since \( P(x) \) is a degree 4 polynomial with a root of multiplicity 2 at \( x = 2 \), and roots of multiplicity 1 at \( x = 0 \) and \( x=-2 \), we can write the polynomial in factored form as:
\( P(x)=a(x - 2)^2(x-0)(x + 2) \), where \( a \) is a non - zero constant.
Simplify the expression:
First, \( (x - 2)^2(x)(x + 2)=x(x + 2)(x - 2)^2 \). Using the difference of squares formula \( (a+b)(a - b)=a^{2}-b^{2} \), we know that \( (x + 2)(x - 2)=x^{2}-4 \), so \( x(x^{2}-4)(x - 2)=x(x^{3}-2x^{2}-4x + 8)=x^{4}-2x^{3}-4x^{2}+8x \). But we can also simplify the factored form as \( P(x)=a x(x + 2)(x - 2)^2=a x(x^{2}-4)(x - 2)=a(x^{3}-4x)(x - 2)=a(x^{4}-2x^{3}-4x^{2}+8x) \)

Step2: Substitute the point \((5,94.5)\) into the polynomial to find \( a \)

We know that the polynomial passes through the point \( (5,94.5) \), which means when \( x = 5 \), \( P(5)=94.5 \)
Substitute \( x = 5 \) into \( P(x)=a x(x + 2)(x - 2)^2 \):
\( P(5)=a\times5\times(5 + 2)\times(5 - 2)^2 \)
Calculate \( 5\times(5 + 2)\times(5 - 2)^2=5\times7\times9 = 315 \)
So we have the equation \( 315a=94.5 \)

Step3: Solve for \( a \)

To find \( a \), divide both sides of the equation \( 315a = 94.5 \) by 315:
\( a=\frac{94.5}{315}=0.3=\frac{3}{10} \)

Step4: Write the final form of the polynomial

Substitute \( a=\frac{3}{10} \) back into the factored form of the polynomial:
\( P(x)=\frac{3}{10}x(x + 2)(x - 2)^2 \)
We can also expand it:
First, \( (x - 2)^2=x^{2}-4x + 4 \)
\( x(x + 2)=x^{2}+2x \)
Then \( (x^{2}+2x)(x^{2}-4x + 4)=x^{4}-4x^{3}+4x^{2}+2x^{3}-8x^{2}+8x=x^{4}-2x^{3}-4x^{2}+8x \)
Multiply by \( \frac{3}{10} \): \( P(x)=\frac{3}{10}(x^{4}-2x^{3}-4x^{2}+8x)=\frac{3}{10}x^{4}-\frac{3}{5}x^{3}-\frac{6}{5}x^{2}+\frac{12}{5}x \)
Or in factored form: \( P(x)=0.3x(x + 2)(x - 2)^2 \)

Answer:

\( P(x)=\frac{3}{10}x(x + 2)(x - 2)^2 \) (or expanded form \( P(x)=\frac{3}{10}x^{4}-\frac{3}{5}x^{3}-\frac{6}{5}x^{2}+\frac{12}{5}x \))