14. Find the remainder when [tex]x^3 - Px^2 + 6x - P[/tex] is divided by [tex]x - p[/tex].
15. One of the zeros of the polynomial [tex]2x^2 + 7x - 4[/tex] is:
(a) ... (b) ... (c) -[tex]\frac{1}{2}[/tex] (d) -2
16. Rationalize the denominator of [tex]\frac{1}{\sqrt{3} - \sqrt{2} - 1}[/tex].
17. Find the square root of [tex]7 + 4\sqrt{3}[/tex].
18. Factorize [tex](x^2 + 3x)^2 - 5(x^2 + 3x) - y(x^2 + 3x) + 5y[/tex].
19. If [tex]x = \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}[/tex] and [tex]y = \frac{9}{\sqrt{3} - \sqrt{2}}[/tex], find [tex]x^2 + y^2 + xy[/tex].
20. Find the value of [tex]x^3 + y^3 - 12xy + 64[/tex] when [tex]x + y = -4[/tex].
21. If the polynomials [tex]az^2 + 4z^2 + 3z - 4[/tex] and [tex]z^3 - 4z[/tex] leave the same remainder when divided by [tex]z - 3[/tex], find the value of [tex]a[/tex].
22. Without actual division, prove that [tex]2x^4 - 5x^3 + 2x^2 - x + 2[/tex] is divisible by [tex]x^2 - 3x + 2[/tex].
Answer (1)
Let's address the question regarding the polynomial division:
We need to find the remainder when x 3 − P x 2 + 6 x − P is divided by x − p .
To find the remainder when a polynomial f ( x ) is divided by another binomial x − a , we use the Remainder Theorem. According to this theorem, the remainder of this division is simply f ( a ) .
In this case, the divisor is x − p , which means we substitute x = p into the polynomial x 3 − P x 2 + 6 x − P :
f ( p ) = p 3 − P p 2 + 6 p − P
This expression p 3 − P p 2 + 6 p − P is the remainder when the polynomial x 3 − P x 2 + 6 x − P is divided by x − p . Therefore, the remainder is p 3 − P p 2 + 6 p − P .
Feel free to ask if you have more questions!
