Algebra Overview - 21 Questions

Algebra Questions

1. Binomial Theorem

Expand fully: \((3x-2y)^5\).

\[ (3x-2y)^5=\sum_{k=0}^{5}\binom{5}{k}(3x)^{5-k}(-2y)^k =243x^5-810x^4y+1080x^3y^2-720x^2y^3+240xy^4-32y^5. \]

2. Fractions

Write as a single fraction in simplest form: \[ \frac{x-1-\tfrac{6}{x}}{\,x-\tfrac{4}{x}\,}. \]

Multiply top and bottom by \(x\): \[ \frac{x^2-x-6}{x^2-4}=\frac{(x-3)(x+2)}{(x-2)(x+2)}=\frac{x-3}{x-2}, \quad x\neq 0,\pm2. \]

3. Surds

If \(x=\sqrt a+\tfrac1{\sqrt a}\) and \(y=\sqrt a-\tfrac1{\sqrt a}\) with \(a>0\), find \(x^2-y^2\).

\[ x^2-y^2=(x-y)(x+y)=\Big(\tfrac{2}{\sqrt a}\Big)\big(2\sqrt a\big)=4. \]

4. Making and manipulating formulae

Using the constant-acceleration formula \( s = ut + \tfrac{1}{2} a t^2 \) (with \(t>0\)), rearrange to express \(a\) in terms of \(s, u, t\).

\[ s-ut=\tfrac12 a t^2 \;\Rightarrow\; a=\frac{2(s-ut)}{t^2},\qquad t\neq 0. \]

5. Linear simultaneous equations (two variables)

\(\;3x+2y=9,\quad 2x-y=-1.\)

From \(2x-y=-1\Rightarrow y=2x+1\). Substitute: \(3x+2(2x+1)=9\Rightarrow7x=7\Rightarrow x=1\), \(y=3\).

6. Linear simultaneous equations (three variables)

Let \(f,s,d\) be ages of father, son, daughter. Given \(f=3(s+d)\), \(f+s=9d\), \(f+s+d=40\). Find \(f,s,d\).

From \(f+s=9d\) and \(f+s+d=40\Rightarrow10d=40\Rightarrow d=4\). Then \(f+s=36\) and \(f=3(s+4)=3s+12\Rightarrow 4s+12=36\Rightarrow s=6,\; f=30.\)

7. Solving quadratic equations

Solve \(28=x(31+5x)\) by (i) factors, (ii) completing the square, (iii) quadratic formula.

Rearrange \(5x^2+31x-28=0=(5x-4)(x+7)\Rightarrow x=\frac45,-7\). Completing the square and quadratic formula give the same roots.

8. Quadratic graphs

For \(f(x)=2x^2-6x+11\), find the turning point and solve \(f(x)=10\) from your graph.

\[ f(x)=2\big(x^2-3x\big)+11 =2\Big((x-\tfrac32)^2-\tfrac94\Big)+11 =2(x-\tfrac32)^2+ \tfrac{13}{2}. \] Turning point \(\big(\tfrac32,\tfrac{13}{2}\big)\). \(f(x)=10\Rightarrow 2x^2-6x+1=0\Rightarrow x=\dfrac{3\pm\sqrt7}{2}.\)

9. Nature of quadratic roots

Show roots of \(px^2-(p+q)x+q=0\) are real \(\forall\,p,q\in\mathbb R\) and express them.

\(\Delta=(p+q)^2-4pq=(p-q)^2\ge0\). Roots: \(\displaystyle x=\frac{(p+q)\pm|p-q|}{2p}\) for \(p\neq0\).

10. Linear / non-linear simultaneous equations

\(\;3x-y=1,\quad x^2+4xy=9.\)

\(y=3x-1\Rightarrow x^2+4x(3x-1)=9\Rightarrow 13x^2-4x-9=0\). \(x=\dfrac{4\pm22}{26}\Rightarrow x=1\) or \(-\tfrac{9}{13}\). \(y=2\) or \(-\tfrac{40}{13}\).

11. Rational equations

\(\;\dfrac{1}{x-2}+\dfrac{4}{x+1}=2.\)

\(\dfrac{5x-7}{(x-2)(x+1)}=2\Rightarrow 2x^2-7x+3=0=(2x-1)(x-3)\). \(x=\tfrac12,\,3\) (exclude \(x\neq2,-1\)).

12. Irrational equations

\(\;\sqrt{2x+1}-\sqrt{x-3}=2,\; x\ge3.\)

Square twice: \(x=4\sqrt{x-3}\Rightarrow x^2-16x+48=0\Rightarrow x=4,12\). Both satisfy the original equation.

13. Identities

If \((x+a)^2-(x+b)^2=12x+12\;\forall x\), find \(a,b\).

Difference of squares: \((a-b)(2x+a+b)=12x+12\). \(2(a-b)=12\Rightarrow a-b=6\); \( (a-b)(a+b)=12\Rightarrow a+b=2\). Hence \(a=4,\; b=-2\).

14. Factor Theorem (cubics)

\(f(x)=x^3+ax^2-7x+b\) with factors \(x-1\) and \(x-2\).

\(f(1)=0\Rightarrow a+b-6=0\). \(f(2)=0\Rightarrow 4a-6+b=0\Rightarrow a=0,\;b=6\). \(f(x)=x^3-7x+6=(x-1)(x-2)(x+3)\).

15. Quadratic factor of a cubic

If \(x^2-px+1\) divides \(ax^3+bx+c\;(a\ne0)\), show \(c^2=a(a-b)\).

Write \(ax^3+bx+c=(x^2-px+1)(Ax+B)\). Compare: \(A=a,\;B=ap,\;b=a- ap^2\), \(c=B=ap\). Thus \(c^2=a^2p^2=a\big(a-ap^2\big)=a(a-b).\)

16. Graphing polynomial curves

The graph of \(y=f(x)\) (degree \(4\)) is shown. Given that the curve passes through \((0,-54)\) and has \(x\)-intercepts at \(-3\), \(1\), and \(3\) (with \(x=3\) a double root), find an expression for \(f(x)\).

From the intercepts and multiplicity, write \[ f(x)=k(x+3)(x-1)(x-3)^2. \] Use the point \((0,-54)\): \[ f(0)=-54=k(3)(-1)(9)=-27k \;\Rightarrow\; k=2. \] Hence a correct expression is \[ \boxed{\,f(x)=2(x+3)(x-1)(x-3)^2\,}. \] Expanded form (optional): \[ f(x)=2x^4-8x^3-12x^2+72x-54. \] Check: \(f(0)=-54\) and the roots are \(-3,1,3,3\) as required.

17. Modulus inequalities

Solve \(|2x+5|<3\).

\(-3<2x+5<3\Rightarrow -8<2x<-2\Rightarrow -4<x<-1.\)

18. Abstract inequalities

Prove \(x^2+y^2\ge\frac12(x+y)^2\) for \(x,y\in\mathbb R\).

Since \((x-y)^2\ge0\Rightarrow x^2+y^2\ge2xy\). Hence \(x^2+y^2\ge \tfrac12(x^2+2xy+y^2)=\tfrac12(x+y)^2.\)

19. Rational inequalities

Solve \(\dfrac{5-x}{x-2}<1,\;x\neq2.\)

\(\dfrac{7-2x}{x-2}<0\). Critical points \(x=2,\;\tfrac{7}{2}\). Sign chart gives \(2<x<\tfrac{7}{2}\).

20. Logs and log equations

\(\;\log_5 x = 1 + \log_2\!\big(\frac{3}{2x-1}\big),\;x>\tfrac12.\)

Write \(1=\log_2 2\Rightarrow \log_5 x=\log_2\!\big(\tfrac{6}{2x-1}\big)\). Change of base: \(\dfrac{\ln x}{\ln 5}=\dfrac{\ln\!\big(\tfrac{6}{2x-1}\big)}{\ln 2}\). Solve numerically (no simple closed form): \(x\approx 2.51619\) (3 s.f.: \(2.52\)).