Download Topic 7 - Polynomials

Education Resources
Polynomials
Higher Mathematics Supplementary Resources
Section A
This section is designed to provide examples which develop routine skills necessary
for completion of this section.
R1
I can solve cubic equations.
1.
Factorise and solve:
(a)
(b)
(b)
(c)
(d)
(e)
(f)
2.
Factorise and solve:
(a)
(b)
(c)
(d)
Higher Mathematics – Polynomials
Page 1
R2
Approximate roots.
1.
(a) Show that a solution to the equation
between 0 and 1.
lies
(b)
Find the solution correct to two decimal places.
(a)
Show that this equation has roots between:
2.
(i) -2 and -1
3.
(ii) 1 and 2
(iii) 2 and 3
(b)
Find the root that lies between 1 and 2 correct to 2 decimal places.
(c)
Find the root that lies between 2 and 3 correct to 1 decimal place.
The equation
has only one root.
(a)
Between what two consecutive whole numbers does the root lie?
(b)
Find the root correct to 1 decimal place.
SLC Education Resources – Biggar High School
Page 2
Section B
This section is designed to provide examples which develop Course Assessment
level skills
NR1
I can factorise a polynomial expression using the factor theorem.
1.
Show that
is a root of
Hence factorise
2.
3.
4.
5.
fully.
(a)
Show that
(b)
Hence factorise
(a)
Show that
(b)
Hence factorise
Show that
(a)
Factorise
(b)
Solve
.
is a factor of
fully.
is a factor of
fully.
is a factor of
.
fully
–
Factorise fully
a)
b)
c)
d)
e)
f)
Higher Mathematics – Polynomials
Page 3
NR2
I can evaluate an unknown coefficient of a polynomial by applying the
remainder and/or the factor theorem.
1.
Given that
is a factor of
, and the remainder when
is 54, find the values of a and b.
2. Find
3.
4.
if
is a factor of
is divided by
.
Given that (x + 1) is a factor of
(a) Find the value of p
(b) Hence or otherwise, solve
Find the value of k if (x+5) is a factor of
5. Given that (x-1) is a factor of
factorise fully.
6. Given that x = 3 is a root of the equation
7. When
, find the value of ‘t’ and hence
, find p.
is divided by (x+3) the remainder is 16.
Find the value of p.
SLC Education Resources – Biggar High School
Page 4
NR3
I can solve a polynomial equation to determine where a curve cuts the
x-axis.
1.
A function is defined on the set of real numbers by
Find where the graph of y =
2.
(a)
the x -axis;
(b)
the y -axis.
cuts:
(Non-calculator)
A function is defined by the formula
is a real number.
is a factor of
where x
(a)
Show that
(b)
Find the coordinates of the points where the curve with
equation
3.
.
and hence factorise
crosses the x and y -axes.
A function is defined by the formula
fully.
(Non-calculator)
.
Find the exact values where the graph of y =
meets the x and y -axes.
(Non-calculator)
4.
(a)
(b)
(i)
Show that
is a factor of
(ii)
Hence or otherwise factorise
.
fully.
One of the turning points of the graph of y =
Write down the coordinates of this turning point.
5.
Find where the graph of
x and y -axes.
Higher Mathematics – Polynomials
lies on the x -axis.
(Non-calculator)
meets the
(Non-calculator)
Page 5
NR4
I can find points of intersection by solving polynomial equations.
1.
C
B
A
Find the coordinates of the points of intersection namely A, B and C where the line
meets the graph
. (NB the diagram is not drawn to scale)
2.
C
B
-1.5
1.5
A
Find the coordinates A, B and C of the points of intersection between the line
curve
.
SLC Education Resources – Biggar High School
and the
Page 6
3.
( NB diagram not to scale)
10
C
B
2
-1
3
4
A
a) Fully factorise the polynomial
b) Hence or otherwise find the coordinates of the points of intersection of the line
and the graph
4.
18
A
B
C
-3
Find the points of intersection of the line
Higher Mathematics – Polynomials
2
and the graph of
.
Page 7
NR5
1.
I have experience of cross topic exam standard questions.
Given that
a) Formula for
b) Hence factorise
c) Solve
2. (a)
(i) Show that
(ii) Factorise
(iii) Solve
and
is a factor of
fully
is a factor of
fully.
.
.
b)The diagram shows the curve with equation
A
0
, find
.
B
C
The curve crosses the x-axis at A, B and C.
Determine the shaded area.
3. The diagram shows a sketch of the graph of
-1
0
Q
a) Show that the graph cuts the x-axis at (-1, 0).
b) Hence or otherwise fine the coordinates of Q.
c) Find the shaded area.
SLC Education Resources – Biggar High School
Page 8
4. Functions
are defined on the set of real numbers by



a) Find
b) Show that
c) (i) Show that (
is a factor of
(ii) Factorise
d) Hence or otherwise solve
5. The line with equation y = 5x – 3 is a tangent to the curve with equation
at A (1,2), as shown on the diagram. Show that the point B is (-3,-18)
A (1,2)
0
B
Higher Mathematics – Polynomials
Page 9
6.
16
B
A
a)
b)
c)
d)
Find where the graph
crosses the axis.
Calculate the points of intersection of the line
and the graph.
Find the shaded area A
Find the shaded area B
SLC Education Resources – Biggar High School
Page 10
Answers
R1
1. (a) x = 3, x = 1(repeated) (b) x = 1, x = 2, x = 3
(c) x = -3, x = 1,x = 2
(d) x = -2, x = 1(repeated)
(e) x = -1, x = 0.5, x = -0.5 (f) x = -3, x = -1,x = 0.5
2. (a) x = -0.5, x = 0.5, x = 1 (b) x = 2 (repeated)
(c) x = -2, x = -0.5, x = 1/3
R2
1. (a) proof
2. (a) proof
3. (a) 2 and 3
(b) 0.47
(b) 1.54
(b) 2.2
(c) 2.68
NR1
1.
2.
3.
4. (a)
(b)
,
5. (a)
(b)
(c)
(d)
(e)
(f)
Higher Mathematics – Polynomials
Page 11
NR2
1. a = -10, b =10
2. p = 27
3. (a) p = -5
(b) x = -1, , -2
4. k = 2
5. (a)t = -3
(b) (x-1)(x2 +2x+4)
6. p = /3
7. p = 47
NR3
1.
(a)
(–3, 0)
(b)
(0, 12)
2.
(a)
proof
(b)
(–2, 0), ( , 0), (5, 0); (0, 10)
3.
x = 0, –
4.
(a)
(ii)
(b)
5.
(–6, 0), (–2, 0) repeated, (4, 0)
,
(i) proof
(3, 0)
NR4
1.
A(-3,-5)
B(0,-2) C(2,0)
2.
A(-2,-1)
B(-1,1) C(1,5)
3.
A(-0.5,-4)
B(2,1) C(4,5)
4.
A(-4,6)
B(-2,4) C(2,0)
NR5
1. (a)
(b) (x-8) (x+8) (2x+1)
(c) x = 8, x= -8, x= -1/2
SLC Education Resources – Biggar High School
Page 12
2. (a)( i) proof
(ii) (x-2) (x-4) (x+1)
(iii) x = 2, x = 4, x = -1
(b) 61/12
3. (a) proof
(b) (2,0)
(c) 45/11
4. (a) 2x3 +3
(b) Proof
(d) x=2, x=0.5, x=3
(c) (x-2)(2x-1)(x-3)
5. proof
6. (a) (-4,0) (-2,0) (2,0)
(c)
Higher Mathematics – Polynomials
(b)
(1,15) (-1,9) (-4,0)
(d)
Page 13