1. Identify the degree, end behavior and roots (with...

1. Identify the degree, end behavior and roots (with multiplicities) of the following polynomials. (a) f(x) = x²(x-1)³(x²+4)(x²-4)(3x-4) (b) f(x) = 4x³(x²-9)²(x²-2)(x+6)? 2. Use the following information to write out the polynomial functions in factored form. Sketch a graph. (a) Degree: 3; has roots: -2,1 (multiplicity 2); leading coefficient: -2

Transcript

00:01 Hi, here for the given question in the first part we need to find the value of degree in behavior and roots of the given polynomial.

00:09 So the first polynomial is f of x is equal to x square multiplied with x minus 1 cube multiplied with x square plus 4 multiplied with x square minus 4 multiplied with 3x minus 4.

00:21 So here this is our polynomial.

00:23 Now here we need to find the value of degree.

00:26 So here for the degree which is highest power will be 2 plus 3 plus 2 plus 2 plus 1.

00:32 So this is equal to 10.

00:34 So here degree of the polynomial is 10 and here for the end behavior we know that we need to check the behavior of x to the power 10.

00:44 We know that when x tends to positive infinity here the value of f of x tends towards positive infinity and when x towards the negative infinity we have f of x towards the negative infinity and further here in our case we need to write about the roots.

01:01 So here for the roots of the polynomial we will consider the different parts of the polynomial.

01:07 So x square which implies root equals to 0 x minus 1 whole cube which implies root is equal to plus 1 and the value of multiplicity is equal to 3 and here we have x square plus 4 which implies the value of root equals to plus or minus 2 iota and x square minus 4 implies root is equal to plus or minus 2 and further 3x minus 4 implies x is equal to 4 by 3.

01:37 So here this is the solution for the first part.

01:39 Now for the second part we are given polynomial f of x is equal to 4x cube multiplied with x square minus 9 whole square multiplied with x square minus 2 into x plus 6 to the power 7.

01:52 So here in our case now we need to write down about the degree.

01:55 Degree is the highest power.

01:57 So we have 3 plus 2 into 2 plus 2 plus 7.

02:03 So here this is equal to 16 degree polynomial.

02:07 Now further we know about the end behavior.

02:09 So here the end behavior is completely dependent on the value of x.

02:14 If x tends towards positive infinity f of x tends towards positive infinity if x tends towards negative infinity we have f of x towards negative infinity.

02:25 Now further we need to write about the roots.

02:28 So here also the value of the roots can be determined by splitting the equation.

02:33 So here we have 4x cube...

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