Given that is a factor of
, find the value of the constant
.
You must make your method clear.
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Given that is a factor of
, find the value of the constant
.
You must make your method clear.
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Given that is a factor of
, find the value of the constant
.
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Expand and simplify
(i)
(ii)
(iii)
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Use the factor theorem to verify that is a factor of
.
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Factorise
(i)
(ii)
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Divide by
.
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Given has a root at
, fully factorise
.
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Use polynomial division to show that is a factor of
.
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Given is a root of the function
, fully factorise
.
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Given that is a factor of
, find the value of
.
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Given that is a root of the function
, find the possible values of
.
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Given that is a factor of
, fully factorise
.
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Sketch the graph of , labelling the coordinates of all points where the graph intersects the coordinate axes.
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Find the remainder when is divided by
.
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The function is given by
, where
and
are constants.
Given that both and
are factors of
find the values of
and
.
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Use the factor theorem to show that is a factor of
.
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Find the constants ,
and
such that
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Hence show that has only one real root.
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Write down the real root of the equation
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where
is a constant
Given that is a factor of
, find the value of
.
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Expand and simplify .
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A rectangle has side lengths of units and
units. Find an expression for the area of the rectangle in terms of
and
.
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Given that , where
are constants, find the values of
.
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Factorise completely .
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Divide by
.
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Find the remainder when is divided by
.
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Given that is a factor of
, factorise
completely.
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Show that where
are constants to be found.
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Hence factorise completely.
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Write down all the real roots of the equation .
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Use the factor theorem to show that is a factor of
.
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Factorise completely.
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Write down all the real roots of the equation .
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. Given that
and
:
find the values of and
.
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Factorise completely.
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In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
where is a positive constant.
Given is a factor of
, show that
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Hence
(i) Find the value of
(ii) use algebra to find the exact solutions of the equation
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(i) Calculate
(ii) Write as a product of two algebraic factors.
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Using the answer to (a)(ii), prove that there are exactly two real solutions to the equation
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Expand and simplify .
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A square has side lengths of units. Find an expression for the length of the diagonal of the square in terms of
and
.
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Given that , where
are constants, find the values of
and
.
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Factorise completely .
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Divide by
.
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Find the remainder when is divided by
.
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Given that is a factor of
, factorise
completely.
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Show that where
and
are constants to be found.
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Hence factorise completely.
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Write down all the real roots of the equation .
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Use the factor theorem to show that is a factor of
.
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Factorise completely.
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Write down all the real roots of the equation .
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. Given that
and
:
find the values of and
.
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Factorise completely.
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Figure 4 shows a sketch of part of the curve with equation
and part of the curve with equation
Verify that the curves intersect at
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The curves intersect again at the point
Using algebra and showing all stages of working, find the exact coordinate of
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In this question you must show detailed reasoning.
Solutions relying on calculator technology are not acceptable.
The curve has equation
The curve has equation
Verify that when the curves
and
intersect.
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The curves also intersect when .
Given that
use algebra to find the exact value of .
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Expand and simplify .
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A cuboid has a length of units, a width of
units, and a height of
units. Find an expression for the volume of the cuboid in terms of
and
.
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Given that , where
and
are constants, find the values of
and
.
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Factorise completely .
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Divide by
.
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Find the remainder when is divided by
.
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Given that is a factor of
, factorise
completely.
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Show that where
and
are constants to be found.
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Given that is a factor of
, factorise
completely.
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Hence show that the equation has exactly 2 real roots.
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Given that 3 is a root of the equation , prove that the equation has no other real roots.
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Show that and
.
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Hence, solve .
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