5.03 Polynomial theorems
Simplify the following:
\dfrac{\left( n^{7} r^{2}\right)^{4}}{\left( n^{4} r\right)^{4}}
\dfrac{9 x^{5} + 8 x^{4}}{x}
\dfrac{40 x^{3} - 48 x^{2} + 32}{8}
\dfrac{15 x^{4} - 24 x^{3} - 21 x^{2} + 12 x}{3 x}
Complete the following statement:
\dfrac{⬚ x^{7} - ⬚ x^{5}}{2 x^{⬚}} = 3 x^{4} - 4 x^{2}
What polynomial, when divided by 2 x^{2}, produces 8 x^{6} - 5 x^{4} + 6 x^{2} as a quotient?
For each of the following, find the remainder when P \left( x \right) is divided by A \left( x \right):
P \left( x \right) = 3 x^{4} - 3 x^{3} - 5 x^{2} + 4 x - 5, A \left( x \right) = x + 5
P \left( x \right) = 3 x^{4} + 5 x^{3} - 2 x^{2} + 6 x + 7, A \left( x \right) = 3 x + 1
P \left( x \right) = 2x^4-3x^3+6x^2-10, A \left( x \right) = x -1
P \left( x \right) = x^3-2x^2+9x-1, A \left( x \right) = x-4
P \left( x \right) = 4x^5-6x^3-7x^2+9x, A \left( x \right) = 2x-5
P \left( x \right) = 6x^4-x^3+9x^2+10x-8, A \left( x \right) = 4x+2
Find the value of k for each of the following:
The remainder when 3 x^{3} + 4 x^{2} + 4 x + k is divided by x - 2 is 52.
The remainder when 4 x^{3} - 2 x^{2} + k x-1 is divided by x - 2 is 15.
Write down all the possible rational zeros of the following polynomials:
P \left( x \right) = 3 x^{4} - 3 x^{3} - 2 x^{2} + 5 x + 6
P \left( x \right) = 6 x^{4} + 9 x^{3} + 2 x^{2} + 5 x + 4
P \left( x \right) = x^{3} + 4 x^{2} - 7 x - 10
Is \dfrac{2}{5} a possible rational zero of P \left( x \right) = 5 x^{3} - 4 x^{2} - 7 x + 10?
The polynomials P \left( x \right) = x^{3} + 2 x^{2} - 5 x + n and Q \left( x \right) = x^{3} + 4 x - 11 give the same remainder when divided by x - 4. Solve for n.
Consider the polynomials P \left( x \right) = x^{4} - 5 x^{3} - k x + m and Q \left( x \right) = k x^{2} + m x - 5. The remainder when P \left( x \right) is divided by x + 2 is 53, while the remainder when Q \left( x \right) is divided by x + 2 is - 31.
Solve for k.
Solve for m.
Hence, find the remainder when P \left( x \right) is divided by x - 4.
Write the following polynomials as a product of linear factors:
x^{3} - 6 x^{2} + 11 x - 6
4 x^{3} - x^{2} - 29 x + 30
Consider the polynomial P \left( x \right) = x^{3} - 4 x^{2} - 11 x + 30.
Write down all the possible zeros.
Find the value of P \left( - 1 \right).
Find the value of P \left( - 3 \right).
Find the value of P \left( 2 \right).
Factorise P \left( x \right) = x^{3} - 4 x^{2} - 11 x + 30.
Consider the division \dfrac{4 x^{2} - 3 x - 6}{x - 2}.
Find the remainder.
Is x - 2 a factor of P \left( x \right)?
Consider the division \dfrac{x^{3} - 5 x^{2} - 2 x - 1}{x + 1}.
Find the remainder.
Is x + 1 a factor of P \left( x \right) ?
Consider the division \dfrac{x^{2} + 4 x - 32}{x + 8}.
Find the remainder.
Is x + 8 a factor of P \left( x \right)?
Factorise x^{2} + 4 x - 32.
Show that x + 2 is a factor of P \left( x \right) = x^{4} + 7 x^{3} + 8 x^{2} - 28 x - 48.
Consider \left( 4 x^{3} + 20 x^{2} + 3 x + 15\right) \div \left(x + 5\right).
Show that x + 5 is a factor of P \left( x \right).
Factorise 4 x^{3} + 20 x^{2} + 3 x + 15.
Consider \left(12 + 9 x + x^{2} - x^{3}\right) \div \left(4 - x\right).
Show that 4 - x is a factor of P \left( x \right).
Factorise 12 + 9 x + x^{2} - x^{3}.
The polynomial P \left( x \right) = x^{3} + a x^{2} + b - 10 x is divisible by both x+1 and x+2.
Solve for the value of a.
Hence, solve for the value of b.
The polynomials 4 x^{2} - 13 x - 12 and 5 x^{2} + 11 x + k have a common factor of x + p, where p is an integer.
Solve for p.
Hence, solve for k.
The polynomial 3 x^{3} + p x^{2} + q x + 2 has a factor of x + 1, but when divided by x - 1, it leaves a remainder of 24.
Solve for p.
Solve for q.
Hence, factorise the polynomial completely.
The polynomial P \left( x \right) = x^{3} + a x^{2} + b + 40 x is divisible by both x+3 and x+4.
Solve for a.
Hence, solve for b.
The polynomial 3 x^{3} + p x^{2} + 8 x + q is divisible by x^{2} - 7 x + 12.
Solve for p.
Solve for q.
Hence, factorise the cubic completely.
The polynomial Q \left( x \right) = x^{4} + 8 x^{3} + a x^{2} - 74 x + b is divisible by both x - 3 and x + 5.
Solve for a.
Solve for b.
Hence, factorise Q \left( x \right) completely.
Consider the rectangle shown with an area of (2 x^{4} - 8 x)\text{ units}^2:
Find a polynomial expression for its length.
Consider the rectangle shown with an area of (21 x^{3} + 6 x^{2} - 15 x - 9)\text{ units}^2:
Find a polynomial expression for its length.
Consider the triangle shown with the given area of (19 n^{3} + 13 n^{2} + 11 n)\text{ units}^2.
Find a polynomial expression for its height.
