Let's solve the given problem step by step.
### Step 1: Given Information and Equations
We are given a quadratic polynomial:
[tex]\[ 2x^2 + px + 4 \][/tex]
We know that one zero (or root) of the polynomial is 2. We need to find the other zero as well as the value of [tex]\( p \)[/tex].
### Step 2: Using the Zero of the Polynomial
Since 2 is a zero of the polynomial, it satisfies the polynomial equation:
[tex]\[ 2(2)^2 + p(2) + 4 = 0 \][/tex]
### Step 3: Simplifying the Equation
Substitute [tex]\( x = 2 \)[/tex] into the polynomial:
[tex]\[ 2(2)^2 + p(2) + 4 = 0 \][/tex]
Simplify the quadratic term and constant:
[tex]\[ 2 \cdot 4 + 2p + 4 = 0 \][/tex]
[tex]\[ 8 + 2p + 4 = 0 \][/tex]
Combine the constants:
[tex]\[ 2p + 12 = 0 \][/tex]
### Step 4: Solving for [tex]\( p \)[/tex]
Isolate [tex]\( 2p \)[/tex]:
[tex]\[ 2p = -12 \][/tex]
Divide by 2:
[tex]\[ p = -6 \][/tex]
Thus, the value of [tex]\( p \)[/tex] is:
[tex]\[ p = -6 \][/tex]
### Step 5: Finding the Other Zero
Now that we have the value of [tex]\( p \)[/tex], let's use the relationship between the roots and the coefficients of a quadratic polynomial.
For a quadratic polynomial [tex]\( ax^2 + bx + c \)[/tex] with roots [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex]:
- The sum of the roots (α + β) is given by [tex]\( -\frac{b}{a} \)[/tex]
- The product of the roots (αβ) is given by [tex]\( \frac{c}{a} \)[/tex]
In our polynomial [tex]\( 2x^2 - 6x + 4 \)[/tex]:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = -6 \)[/tex]
- [tex]\( c = 4 \)[/tex]
### Step 6: Sum and Product of the Roots
The sum of the roots [tex]\( α + β \)[/tex] is:
[tex]\[ α + β = -\frac{b}{a} = -\frac{-6}{2} = 3 \][/tex]
The product of the roots [tex]\( αβ \)[/tex] is:
[tex]\[ αβ = \frac{c}{a} = \frac{4}{2} = 2 \][/tex]
### Step 7: Using Known Root to Find the Other Root
We know one root [tex]\( α = 2 \)[/tex]. Let [tex]\( β \)[/tex] be the other root:
[tex]\[ 2 + β = 3 \][/tex]
Solve for [tex]\( β \)[/tex]:
[tex]\[ β = 3 - 2 = 1 \][/tex]
Thus, the other zero is:
[tex]\[ β = 1 \][/tex]
### Final Answer
- The value of [tex]\( p \)[/tex] is [tex]\( -6 \)[/tex].
- The other zero of the polynomial is [tex]\( 1 \)[/tex].
