Write the polynomial function as a product of linear factors.
6x^4-7x^3-10x^2+17x-6
To factor the given polynomial, 6x^4 - 7x^3 - 10x^2 + 17x - 6, we can use the Rational Root Theorem to find any potential rational roots, and then divide the polynomial by those roots to obtain the linear factors.
The Rational Root Theorem states that if a polynomial has a rational root p/q (where p is a factor of the constant term and q is a factor of the leading coefficient), then p is a factor of the constant term and q is a factor of the leading coefficient.
The constant term is -6, and its factors are ±1, ±2, ±3, and ±6.
The leading coefficient is 6, and its factors are ±1, ±2, ±3, and ±6.
By trying out the possible combinations of p/q, we find that x = 1 is a root of the polynomial since (1)^3 - 7(1)^2 - 10(1) + 17(1) - 6 = 0.
Now, we can use synthetic division or long division to factor out the root x = 1. Dividing the polynomial by x - 1:
1 │ 6 - 7 + 0 - 10 + 17 - 6
│ 6 - 1 - 1 - 9 + 8 + 25
└───────────────────────
6 - 1 - 1 - 9 + 8 + 19
After the division, we are left with the quotient 6x^3 - x^2 - x - 9 + 8x + 19. The remainder is 0, indicating that the division is exact.
Now, we continue to find the factors of this quadratic polynomial, 6x^3 - x^2 - x - 9 + 8x + 19. Unfortunately, this polynomial does not have any rational roots.
At this point, we can use a factoring technique such as grouping or the quadratic formula to factor the remaining quadratic polynomial.
However, given that the quadratic term does not have any rational roots, the polynomial cannot be factored further into linear factors over the set of rational numbers.
Therefore, the factored form of the polynomial 6x^4 - 7x^3 - 10x^2 + 17x - 6 is (x - 1)(6x^3 - x^2 - x - 9 + 8x + 19).
To write the given polynomial function as a product of linear factors, we need to factor it completely. Here's how you can do it step by step:
Step 1: Check for Rational Roots
The Rational Root Theorem tells us that any rational root of a polynomial must be a factor of the constant term (in this case -6) divided by a factor of the leading coefficient (in this case 6). The factors of 6 are ±1, ±2, ±3, and ±6, and the factors of -6 are ±1, ±2, ±3, and ±6. Therefore, the possible rational roots are:
±1/1, ±1/2, ±1/3, ±1/6, ±2/1, ±2/2, ±2/3, ±2/6, ±3/1, ±3/2, ±3/3, ±3/6, ±6/1, ±6/2, ±6/3, ±6/6.
Step 2: Test the Possible Roots
Using synthetic division or substituting each possible root and checking if the result is zero, you can determine which ones are actual roots. In this case, we find that x = 1/2 and x = 2 are roots of the polynomial.
Step 3: Divide the Polynomial by the Found Roots
Using synthetic division or long division, divide the polynomial (6x^4 - 7x^3 - 10x^2 + 17x - 6) by the found roots (1/2 and 2). This will give you a quotient.
Step 4: Factor the Quotient
The quotient obtained in the previous step will be a quadratic that can be factored further if possible. Factorize it completely to obtain the linear factors.
Putting it all together:
After following the above steps, you will find that the given polynomial function (6x^4 - 7x^3 - 10x^2 + 17x - 6) can be factored as:
(x - 1/2) * (x - 2) * (2x - 3) * (3x + 1)