Question

Use the factor theorem to find all real zeros and one factor.

f(x) = 2x³-9x² + 13x - 6; x - 1

Answer 1

`x=1, \quad x=(3)/(2), \quad x=2`

Explanation:

Factor Theorem

If f(x) is a polynomial, and f(a) = 0, then (x – a) is a factor of f(x).

If the coefficients in a polynomial add up to 0, then (x - 1) is a factor.

Given polynomial function:

`f(x)=2x^3-9x^2+13x-6`

Sum the coefficients:

`\implies 2-9+13-6=0`

As the sum of the coefficients is zero, (x - 1) is a factor.

Therefore:

`\implies f(x)=(x-1)(ax^2+bx+c)`

Expand the brackets:

`\implies f(x)=ax^3+bx^2+cx-ax^2-bx-c`

`\implies f(x)=ax^3+(b-a)x^2+(c-b)x-c`

Compare the coefficients with the given polynomial.

As the coefficient of the leading term of the polynomial is 2:

`\implies a=2`

As the coefficient of the term in x² is -9:

`\implies b-a=-9`

`\implies b-2=-9`

`\implies b=-7`

As the constant of the given polynomial is -6:

`\implies c=6`

Substitute the found values of a, b and c into the factored form of the polynomial:

`\implies f(x)=(x-1)(2x^2-7x+6)`

Factor the quadratic:

`\implies 2x^2-7x+6`

`\implies 2x^2-4x-3x+6`

`\implies 2x(x-2)-3(x-2)`

`\implies (2x-3)(x-2)`

Therefore, the fully factored polynomial is:

`\implies f(x)=(x-1)(2x-3)(x-2)`

To find the zeros, set the function to zero:

`\implies f(x)=0`

`\implies (x-1)(2x-3)(x-2)=0`

Apply the zero-product property:

`\implies x-1=0 \implies x=1`

`\implies 2x-3=0 \implies x=(3)/(2)`

`\implies x-2=0 \implies x=2`

Therefore, the real zeros of the given polynomial are 1, ³/₂ and 2.

Answer 2

• x = {1, 1.5, 2}

Explanation:

Given

• Polynomial f(x) = 2x³ - 9x² + 13x - 6,
• One of the factors x - 1.

Factorize the polynomial using the given details.

Since one of the factors known, try to factor it out and futher:

• 2x³ - 9x² + 13x - 6 =
• 2x³ - 2x² - 7x² + 7x + 6x - 6 =
• 2x²(x - 1) - 7x(x - 1) + 6(x - 1) =
• (x - 1)(2x² - 7x + 6) =
• (x - 1)(2x² - 3x - 4x + 6) =
• (x - 1)[x(2x - 3) - 2(2x - 3)] =
• (x - 1)(x - 2)(2x - 3)

Find its zero's:

• x - 1 = 0 or x - 2 = 0 or 2x - 3 = 0
• x = 1 or x = 2 or x = 1.5