The polynomial \(\mathrm{p}\) is such that \(\mathrm{p}(x)=3x^3-7x^2+ax+b\), where \(a\) and \(b\) are integers.

It is given that \(\mathrm{p}'(-1)=21\) and that \(x-2\) is a factor of \(\mathrm{p}(x)\).

(a) Find the values of \(a\) and \(b\).

(b) Hence write \(\mathrm{p}(x)\) as a product of linear factors with integer coefficients.

(c) Using your values of \(a\) and \(b\), solve the equation \(3\mathrm{e}^{6y}-7\mathrm{e}^{4y}+a\mathrm{e}^{2y}+b=0\).

The polynomial p is such that \(\mathrm{p}(x)=x^{3}+A x+30\), where \(A\) is a constant. When \(\mathrm{p}(x)\) is divided by \(x+2\) the remainder is 84 . Write \(\mathrm{p}(x)\) as a product of linear factors.

The polynomial \(\mathrm{p}\) is given by \(\mathrm{p}(x)=a^2x^3+2ax^2+ax+2\), where \(a\) is a positive integer.

It is given that \(2x+1\) is a factor of \(\mathrm{p}(x)\).

(a) Find the value of \(a\).

(b) Hence factorise \(\mathrm{p}(x)\).

(c) Hence show that the equation \(\mathrm{p}(x)=0\) has only one real root.

The polynomial p is such that \(\mathrm{p}(x)=5 x^{3}+a x^{2}+39 x+b\), where \(a\) and \(b\) are constants. (a) Given that \(x+3\) is a factor of both \(\mathrm{p}(x)\) and \(\mathrm{p}^{\prime}(x)\), find the values of \(a\) and \(b\).

(b) Hence solve the equation \(\mathrm{p}(x)=0\).

You must show your working.

(c) Hence, using your values for \(a\) and \(b\), solve the equation \(5 \operatorname{cosec}^{3} 2 \theta+a \operatorname{cosec}^{2} 2 \theta+39 \operatorname{cosec} 2 \theta+b=0 \text { for } 0^{\circ} \leqslant \theta \leqslant 360^{\circ} .\)

The polynomial \(p(x)\) is defined by

\(p(x)=2x^3+11x^2+22x+40.\)

(a) Show that \(x=-4\) is a root of the equation \(p(x)=0\).

(b) Factorise \(p(x)\) and show that the equation \(p(x)=0\) has no other real roots.

The polynomial \(p(x)=6x^3+ax^2+6x+b\), where \(a\) and \(b\) are integers, is divisible by \(2x-1\). When \(p(x)\) is divided by \(x-2\), the remainder is \(120\).

(a) Find the values of \(a\) and \(b\).

(b) Hence write down the remainder when \(p(x)\) is divided by \(x\).

(c) Find the value of \(p''(0)\).

The polynomial \(p(x)=6x^3+ax^2-52x+b\) is exactly divisible by \(2x-3\). It is also given that \(p'(1)=4\).

(a) Find the values of \(a\) and \(b\).

(b) Hence factorise \(p(x)\) fully.

The polynomial \(p(x)\) is such that

\(p(x)=ax^3+13x^2+bx+c,\)

where \(a\), \(b\) and \(c\) are integers. It is given that \(p'(0)=-9\).

(a) Show that \(b=-9\).

It is also given that \(3x+2\) is a factor of \(p(x)\) and that when \(p(x)\) is divided by \(x+1\) the remainder is \(6\).

(b) Find the values of \(a\) and \(c\).

(c) Find the quadratic \(q(x)\) such that

\(p(x)=(3x+2)q(x).\)

(d) Hence find \(p(x)\) as a product of linear factors with integer coefficients.

Do not use a calculator in this question.

The polynomial \(\mathrm p(x)=10x^3+ax^2-10x+b\), where \(a\) and \(b\) are integers, is divisible by \(2x+1\). When \(\mathrm p(x)\) is divided by \(x+1\), the remainder is \(-24\).

(a) Find the value of \(a\) and of \(b\).

(b) Find an expression for \(\mathrm p(x)\) as the product of three linear factors.

(c) Write down the remainder when \(\mathrm p(x)\) is divided by \(x\).

(a) It is given that

\(\mathrm f(x)=4x^3-4x^2-15x+18.\)

Find the equation of the normal to the curve \(y=\mathrm f(x)\) at the point where \(x=1\).

(b) Without using a calculator, it is also given that \(x+a\), where \(a\) is an integer, is a factor of \(\mathrm f(x)\). Find \(a\) and hence solve the equation \(\mathrm f(x)=0\).

\(p(x)=ax^3+3x^2+bx-12\)

has a factor of \(2x+1\). When \(p(x)\) is divided by \(x-3\), the remainder is \(105\).

(a) Find the value of \(a\) and of \(b\).

(b) Using your values of \(a\) and \(b\), write \(p(x)\) as a product of \(2x+1\) and a quadratic factor.

(c) Hence solve \(p(x)=0\).

\(p(x)=6x^3+ax^2+12x+b,\)

where \(a\) and \(b\) are integers.

\(p(x)\) has a remainder of \(11\) when divided by \(x-3\) and a remainder of \(-21\) when divided by \(x+1\).

(a) Given that \(p(x)=(x-2)Q(x)\), find \(Q(x)\), a quadratic factor with numerical coefficients.

(b) Hence solve \(p(x)=0\).

Do not use a calculator in this question.

The polynomial \(p(x)\) is given by

\(p(x)=15x^3+22x^2-15x+2.\)

(a) Find the remainder when \(p(x)\) is divided by \(x+1\).

(b)(i) Show that \(x+2\) is a factor of \(p(x)\).

(b)(ii) Write \(p(x)\) as a product of linear factors.

The polynomial

\(\mathrm{p}(x)=ax^3+bx^2-19x+4,\)

where \(a\) and \(b\) are constants, has a factor \(x+4\) and is such that

\(2\mathrm{p}(1)=5\mathrm{p}(0).\)

(a) Show that

\(\mathrm{p}(x)=(x+4)(Ax^2+Bx+C),\)

where \(A\), \(B\) and \(C\) are integers to be found.

(b) Hence factorise \(\mathrm{p}(x)\).

(c) Find the remainder when \(\mathrm{p}'(x)\) is divided by \(x\).

Do not use a calculator in this question.

\(\mathrm{p}(x)=2x^3-3x^2-23x+12.\)

(a) Find the value of \(\mathrm{p}\left(\frac12\right)\).

(b) Write \(\mathrm{p}(x)\) as the product of three linear factors and hence solve \(\mathrm{p}(x)=0\).

The polynomial \(p(x)\) is defined by \(p(x)=2x^3+ax^2+bx-49\), where \(a\) and \(b\) are constants. When \(p'(x)\) is divided by \(x+3\), the remainder is \(-24\).

(i) Show that \(6a-b=78\).

It is now given that \(2x-1\) is a factor of \(p(x)\).

(ii) Find the values of \(a\) and \(b\).

(iii) Express \(p(x)\) in the form \((2x-1)Q(x)\), where \(Q(x)\) is a quadratic expression.

(iv) Hence factorise \(p(x)\) completely.

(i) Given that \(x-2\) is a factor of \(ax^3-12x^2+5x+6\), use the factor theorem to show that \(a=4\).

(ii) Factorise \(4x^3-12x^2+5x+6\) and hence solve \(4x^3-12x^2+5x+6=0\).

Do not use a calculator in this question.

It is given that \(x+4\) is a factor of

\(p(x)=2x^3+3x^2+ax-12.\)

When \(p(x)\) is divided by \(x-1\), the remainder is \(b\).

(i) Show that \(a=-23\) and find the value of the constant \(b\).

(ii) Factorise \(p(x)\) completely and hence state all the solutions of \(p(x)=0\).

Do not use a calculator in this question.

It is given that \(x+4\) is a factor of

\(p(x)=2x^3+3x^2+ax-12.\)

When \(p(x)\) is divided by \(x-1\), the remainder is \(b\).

(i) Show that \(a=-23\) and find the value of the constant \(b\).

(ii) Factorise \(p(x)\) completely and hence state all the solutions of \(p(x)=0\).

The polynomial \(p(x)=ax^3+17x^2+bx-8\) is divisible by \(2x-1\) and has a remainder of \(-35\) when divided by \(x+3\).

(i) By finding the value of each of the constants \(a\) and \(b\), verify that \(a=b\).

Using your values of \(a\) and \(b\),

(ii) find \(p(x)\) in the form \((2x-1)q(x)\), where \(q(x)\) is a quadratic expression.

(iii) factorise \(p(x)\) completely.

(iv) solve \(a\sin^3\theta+17\sin^2\theta+b\sin\theta-8=0\) for \(0^\circ\lt \theta\lt 180^\circ\).

It is given that

\(p(x)=2x^3+ax^2+4x+b,\)

where \(a\) and \(b\) are constants. It is given also that \(2x+1\) is a factor of \(p(x)\) and that when \(p(x)\) is divided by \(x-1\) there is a remainder of \(-12\).

(i) Find the value of \(a\) and of \(b\).

(ii) Using your values of \(a\) and \(b\), write \(p(x)\) in the form \((2x+1)q(x)\), where \(q(x)\) is a quadratic expression.

(iii) Hence find the exact solutions of the equation \(p(x)=0\).

Without using a calculator, factorise the expression

\(10x^3-21x^2+4.\)

The polynomial \(p(x)\) is \(ax^3+bx^2-13x+4\), where \(a\) and \(b\) are integers. Given that \(2x-1\) is a factor of \(p(x)\) and also a factor of \(p'(x)\),

(i) find the value of \(a\) and of \(b\).

Using your values of \(a\) and \(b\),

(ii) find the remainder when \(p(x)\) is divided by \(x+1\).

A polynomial \(p(x)\) is \(ax^3+8x^2+bx+5\), where \(a\) and \(b\) are integers. It is given that \(2x-1\) is a factor of \(p(x)\) and that a remainder of \(-25\) is obtained when \(p(x)\) is divided by \(x+2\).

(i) Find the value of \(a\) and of \(b\).

(ii) Using your values of \(a\) and \(b\), find the exact solutions of \(p(x)=5\).