Download Polynomials (Chapter 4) - Core 1 Revision 1. The polynomial p(x

Polynomials (Chapter 4) – Core 1 Revision
1.
The polynomial
p(x) = x4 – 2x3 + ax2 + bx + 7
leaves a remainder of 4 when divided by (x – 1) and leaves a remainder of 13 when divided by
(x + 2). Find the values of the constants a and b.
(Total 5 marks)
2.
The function f is given by
f(x) = x 3 + ax2 + bx + 6.
When f(x) is divided by (x  1), the remainder is 24.
When f(x) is divided by (x + 2), the remainder is also 24.
Use the Remainder Theorem to find the numerical values of a and b.
(Total 4 marks)
3.
The polynomial
f(x) = 2x3 + px2 + qx – 6
has (x + 3) as a factor.
Also, when f(x) is divided by (x – 2) the remainder is 40.
(a)
Determine the value of p and the value of q.
(4)
(b)
Using the values of p and q from part (a), find the exact values of the roots of the
equation f(x) = 0.
(4)
(Total 8 marks)
4.
The polynomial p(x) is given by p(x) = x3  3x + 2.
(2)
(a)
Find the remainder when p(x)is divided by x + 1.
(2)
(b)
Given that x + 2 is a factor of p(x), express p(x) as a product of linear factors.
(3)
(c)
Simplify the following algebraic fraction as far as possible:
x 3  3x  2
x2 1
(2)
(Total 9 marks)
South Wolds Comprehensive School
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Polynomials (Chapter 4) – Core 1 Revision
5.
The polynomial p(x) is given by
p(x) = x3 + x2 – 10x + 8
(a)
(i)
Using the factor theorem, show that x – 2 is a factor of p(x).
(2)
(ii)
Hence express p(x) as the product of three linear factors.
(3)
(b)
Sketch the curve with equation y = x3 + x2 – 10x + 8, showing the coordinates of the
points where the curve cuts the axes.
(You are not required to calculate the coordinates of the stationary points.)
(4)
(Total 9 marks)
6.
The polynomial f(x) is given by
f(x) = x3 + px2 + x + 54.
where p is a real number. When f(x) is divided by x + 3, the remainder is –3.
Use the Remainder Theorem to find the value of p.
(3)
(Total 3 marks)
7.
It is given that
f (x) = x3 + x2  16x  16.
(a)
(i)
Find the value of f (1).
(1)
(ii)
Hence write down a factor of f (x).
(1)
(b)
Express f (x) as a product of three linear factors.
(4)
(Total 6 marks)
8.
Divide x3 + 2x2 – 5x – 6 by x + 1.
(Total 3 marks)
9.
f(x) = 5x3 + 24x2 + 29x + 2.
(a)
Use the factor theorem to find one factor of f(x)
(2)
(b)
Hence write f(x) in the form
(x + k)(ax2 + bx + c),
giving the value of each of the constants k, a, b, and c.
(4)
(c)
Hence find the exact solutions to the equation f(x) = 0.
(4)
(Total 10 marks)
South Wolds Comprehensive School
2
Polynomials (Chapter 4) – Core 1 Revision
10.
The function f(x) = 3x³ + a²– 6x – 8 has a factor of (x + 2) Find the value of a and completely
factorise f(x).
(Total 7 marks)
11.
The polynomials f(x) and g(x) are defined by
f(x) = x3 + px2 – x + 5
g(x) = x3 – x2 + px +1
where p is a constant. When f(x) and g(x) are divided by x – 2, the remainder is R in each case.
Find the values of p and R
(Total 5 marks)
12.
A function f is given by f(x) = x3 – 4x2 – 3x + 6.
(a)
Use the factor theorem to find a linear factor of f(x).
(2)
(b)
Write f(x) as a product of two factors and hence find the exact solutions of f(x) = 0.
(5)
(Total 7 marks)
13.
The polynomial p(x) is given by
p(x) = x3  4x2  7x + k
where k is a constant.
(a)
(i)
Given that x + 2 is a factor of p(x), show that k = 10.
(2)
(ii)
Express p(x) as the product of three linear factors.
(3)
(b)
Use the Remainder Theorem to find the remainder when p(x) is divided by x  3.
(2)
(c)
Sketch the curve with equation y = x 3  4x 2 7x + 10, indicating the values where the
curve crosses the x-axis and the y-axis. (You are not required to find the coordinates of
the stationary points.)
(4)
(Total 11 marks)
14.
Given that f(x) = x3 – 4x2 – x + 4,
(a)
find f(1) and f(2),
(2)
(b)
factorise f(x) into the product of three linear factors.
(3)
(Total 5 marks)
15.
The cubic polynomial x3 + ax2 + bx + 4, where a and b are constants, has factors x – 2
and x + 1.
Use the factor theorem to find the values of a and b.
(Total 6 marks)
South Wolds Comprehensive School
3
Polynomials (Chapter 4) – Core 1 Revision
16.
The polynomial p(x) is given by p(x) = x3 – 4x2 + 3x.
(a)
Use the Factor Theorem to show that x – 3 is a factor of p(x).
(2)
(b)
Express p(x) as the product of three linear factors.
(2)
(c)
(i)
Use the Remainder Theorem to find the remainder, r, when p(x) is
divided by x – 2.
(2)
(ii)
Using algebraic division, or otherwise, express p(x) in the form
(x – 2)(x2 + ax + b) + r
where a, b and r are constants.
(4)
(Total 10 marks)
17.
(a)
Express 6x3 + 17x2 + 14x + 3 in the form (x + 1)(ax2 + bx + c), stating the values
of a, b and c.
(3)
(b)
Hence solve the equation 6x3 + 17x2 + 14x + 3 = 0.
(3)
(Total 6marks)
South Wolds Comprehensive School
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