03 - Polynomials

St Joseph’s Institution Secondary Four Mathematics TOPIC − Polynomials Name:_____________________________________ ( Q1)

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Solve the following equations for x a) 2x 3 − 9 x 2 + 3 x + 4 = 0 b) x 3 − 3x 2 + 4 = 0 Given that ( x − 2) is a factor of the polynomial x 3 − 4 x 2 + ax + b = 0 , where a and b are constants. It leaves a remainder of − 60 when the polynomial is divided by ( x + 3 ) . i) Find the values of a and of b. ii) Factorise the polynomial completely and hence solve the 3 2 equation ( x + 1) − 4( x + 1) + ( x + 1) + 6 = 0 . Given that x 3 + mx 2 + nx − 6 is exactly divisible by x 2 − x − 6 , find the values of m and of n. For these values of m and of n, solve completely the equation x 3 + mx 2 + nx − 6 = 0 . Find the remainder when 3 x 3 + x + 1 is divided by ( x + 2) . Hence state the number which should be added to 3 x 3 + x + 1 so that the resultant expression is divisible by ( x + 2) . Deduce the quadratic factor of the new expression.

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Solve 2 x 3 + 5 x 2 − 28 x − 15 = 0 and hence solve 2 sin 3 θ − 15 = 28 sinθ − 5 sin 2 θ , where 0° ≤ θ ≤ 360° .

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The expression px 3 + qx 2 − 7 x + q leaves a remainder of R when it is divided by ( x + 1) and a remainder of ( 2R + 4 ) when it is divided by ( x − 2) . i) Show that 10 p + q = 32 . ii)

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Given further than pq = 6 and p > q , find the value of p and of q. Hence, factorise the expression completely.

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k − 13 x at 3 distinct x points. Given that the x-coordinates of one of the points if intersection is − 2 , find the value of k. Hence, find the x-coordinates of the other 2 points of intersection. The curve y = 6 x 2 + 1 intersects another curve y =

The expression x 3 − 8 x 2 + px − 12 and x 3 + 6 x 2 + ( p − 36 ) x − 30 have a common factor ( x + q ) where q is an integer. Find the values of p and of q. Sketch the graph of y = x 3 − 3 x 2 − x − 12 . Explain how you can use this graph to solve the equation x 3 − 3 x 2 − 6 x + 8 = 0 .

Q10) Given that x 2 + 2 x − 3 is a factor of the polynomial 2 x 4 + 3 x 3 + 2ax 2 + 5bx − 6b , find the value of a and of b, and the other quadratic factor of the polynomial. Q11) The expression ax 3 + 2bx 2 − 34 x + 12 is exactly divisible by x − 3 and leaves a remainder of 32 when divided by x + 1. Find the values of a and b. Given also that ax 3 + 2bx 2 − 34 x + 12 is also exactly divisible by 3 x 2 + kx − 2 , find the value of k. 3 2 2 Q12) Given 4 x  9 x  px  5  Ax  x  3  B  x  1  x  1  2 , for all values of x, evaluate the constants A, B and p.

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