Sure, let's solve the problems step-by-step:
### Question 5
We need to compute the value of [tex]\( (67)^2 - (33)^2 \)[/tex].
To simplify this expression, we can use the difference of squares formula, which states:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
Here, [tex]\( a = 67 \)[/tex] and [tex]\( b = 33 \)[/tex].
Using the formula:
[tex]\[ (67)^2 - (33)^2 = (67 - 33)(67 + 33) \][/tex]
First, calculate the difference:
[tex]\[ 67 - 33 = 34 \][/tex]
Next, calculate the sum:
[tex]\[ 67 + 33 = 100 \][/tex]
Now multiply these two results:
[tex]\[ 34 \times 100 = 3400 \][/tex]
So, the value of [tex]\( (67)^2 - (33)^2 \)[/tex] is [tex]\( 3400 \)[/tex].
Therefore, the answer to Question 5 is:
[tex]\[ \boxed{3400} \][/tex]
### Question 6
Given that [tex]\(-1\)[/tex] is a zero of the polynomial [tex]\( p(x) = a x^3 - x^2 + x + 4 \)[/tex], we need to find the value of [tex]\( a \)[/tex].
A zero of a polynomial means that if we substitute [tex]\(-1\)[/tex] into the polynomial, the result should be zero:
[tex]\[ p(-1) = 0 \][/tex]
Substitute [tex]\(-1\)[/tex] into [tex]\( p(x) \)[/tex]:
[tex]\[ p(-1) = a(-1)^3 - (-1)^2 + (-1) + 4 \][/tex]
Simplify step-by-step:
[tex]\[ p(-1) = a(-1) - 1 + (-1) + 4 \][/tex]
[tex]\[ p(-1) = -a - 1 - 1 + 4 \][/tex]
[tex]\[ p(-1) = -a + 2 \][/tex]
We know that:
[tex]\[ p(-1) = 0 \][/tex]
So:
[tex]\[ -a + 2 = 0 \][/tex]
Solve for [tex]\( a \)[/tex]:
[tex]\[ -a + 2 = 0 \][/tex]
[tex]\[ -a = -2 \][/tex]
[tex]\[ a = 2 \][/tex]
Therefore, the value of [tex]\( a \)[/tex] is:
[tex]\[ \boxed{2} \][/tex]
So, combining:
- The value of [tex]\( (67)^2 - (33)^2 \)[/tex] is [tex]\( 3400 \)[/tex].
- If [tex]\(-1\)[/tex] is a zero of the polynomial [tex]\( p(x) = a x^3 - x^2 + x + 4 \)[/tex], the value of [tex]\( a \)[/tex] is [tex]\( 2 \)[/tex].
