Questions: The first polynomial is a factor of the second one. Factor the second polynomial completely. x-3; x^3-5x^2+3x+9

Transcript text: The first polynomial is a factor of the second one. Factor the second polynomial completely. \[ x-3 ; x^{3}-5 x^{2}+3 x+9 \]

Solution

Solution Steps

Solution Approach

To factor the second polynomial completely, we first use the given factor \(x-3\) to perform polynomial division on the polynomial \(x^3 - 5x^2 + 3x + 9\). This will help us find the quotient polynomial. Once we have the quotient, we can check if it can be factored further. The complete factorization will include the given factor and any factors of the quotient.

Step 1: Define the Polynomials

We are given the factor \(x - 3\) and the polynomial \(x^3 - 5x^2 + 3x + 9\). We will use polynomial division to determine if \(x - 3\) is indeed a factor of the polynomial.

Step 2: Perform Polynomial Division

We divide the polynomial \(x^3 - 5x^2 + 3x + 9\) by the factor \(x - 3\). The result of this division gives us a quotient polynomial and a remainder.

Step 3: Check the Remainder

After performing the division, we find that the remainder is \(0\). This confirms that \(x - 3\) is a factor of the polynomial.

Step 4: Factor the Quotient

Next, we take the quotient obtained from the division and check if it can be factored further. The quotient polynomial is \(x^2 - 2x - 3\). We can factor this polynomial as follows: \[ x^2 - 2x - 3 = (x - 3)(x + 1) \]

Step 5: Combine All Factors

Now, we can express the original polynomial \(x^3 - 5x^2 + 3x + 9\) in its completely factored form: \[ x^3 - 5x^2 + 3x + 9 = (x - 3)(x - 3)(x + 1) = (x - 3)^2(x + 1) \]