Sure, let's find the other two roots of the polynomial [tex]\(4x^3 + 3x^2 - 4x - 3\)[/tex] given that [tex]\(1\)[/tex] is a root.
### Step-by-Step Solution:
#### Step 1: Verify that 1 is a root
Given that 1 is a root, it means:
[tex]\[ P(1) = 4(1)^3 + 3(1)^2 - 4(1) - 3 = 4 + 3 - 4 - 3 = 0 \][/tex]
So, [tex]\( P(1) = 0 \)[/tex].
#### Step 2: Factorize the polynomial using the root
If [tex]\(1\)[/tex] is a root of the polynomial, then [tex]\((x - 1)\)[/tex] is a factor of [tex]\(P(x)\)[/tex]. So, we can write the polynomial as:
[tex]\[ P(x) = (x - 1)Q(x) \][/tex]
Where [tex]\(Q(x)\)[/tex] is a quadratic polynomial.
#### Step 3: Perform polynomial division
We need to divide [tex]\(4x^3 + 3x^2 - 4x - 3\)[/tex] by [tex]\((x - 1)\)[/tex] to find [tex]\(Q(x)\)[/tex].
#### Step 4: Find the quotient polynomial Q(x)
By dividing [tex]\(4x^3 + 3x^2 - 4x - 3\)[/tex] by [tex]\((x - 1)\)[/tex], we get:
[tex]\[ Q(x) = 4x^2 + 7x + 3 \][/tex]
So the polynomial can be expressed as:
[tex]\[ 4x^3 + 3x^2 - 4x - 3 = (x - 1)(4x^2 + 7x + 3) \][/tex]
#### Step 5: Solve for the roots of the quadratic polynomial
Now, we need to solve the quadratic equation:
[tex]\[ 4x^2 + 7x + 3 = 0 \][/tex]
We can use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Where [tex]\(a = 4\)[/tex], [tex]\(b = 7\)[/tex], and [tex]\(c = 3\)[/tex].
Calculate the discriminant:
[tex]\[ b^2 - 4ac = 7^2 - 4(4)(3) = 49 - 48 = 1 \][/tex]
Now find the roots:
[tex]\[ x = \frac{-7 \pm \sqrt{1}}{8} \][/tex]
[tex]\[ x = \frac{-7 \pm 1}{8} \][/tex]
So we get two solutions:
[tex]\[ x = \frac{-7 + 1}{8} = \frac{-6}{8} = -\frac{3}{4} \][/tex]
[tex]\[ x = \frac{-7 - 1}{8} = \frac{-8}{8} = -1 \][/tex]
Therefore, the other two roots of the polynomial [tex]\(4x^3 + 3x^2 - 4x - 3\)[/tex] are:
[tex]\[ x = -\frac{3}{4} \][/tex]
[tex]\[ x = -1 \][/tex]
So, the other two roots are [tex]\(-\frac{3}{4}\)[/tex] and [tex]\(-1\)[/tex].
