Understanding Engineering Mathematics

Complications can arise with such systems of equations, depending on the coefficients a,b,c,d,e,f. For example consider the syst ...

No simple cancellation jumps out at us here. If we want to eliminatey, say, then the easiest way is to multiply (i) by 2 to get ...

where theA,B,Care ‘unknown’. In fact, the only way the left- and right-hand sides can beidenticalis if the coefficients of corre ...

2.2.6 Roots and factors of a polynomial ➤ 39 76 ➤ Shortly, we will be looking at more powerful methods for factorising polynomia ...

Example p(x)≡x^4 + 2 x^3 −x− 2 The roots must be factors of−2 and so the only integer roots are likely to be±1or±2. Wefindbytria ...

are, of course, simply the roots of the polynomial on the left-hand side, by definition, and so the solutions arex=1 (twice) and ...

You may think that the fraction x x does exist, and that it is equal to 1, on cancellation of thex. But again this is only true ...

2.2.8 Algebra of rational functions ➤ 39 76 ➤ Rational functions often cause a lot of problems for beginners in mathematics. The ...

Just as we can simplify the fraction^24 to its ‘lowest’ form^12 , so we may be able to ‘cancel down’ algebraic fractions – but w ...

So, for example, 1 a + 1 b = b ab + a ab = b+a ab = a+b ab This is calledputting the fractions over a common denominator.Itiswha ...

= 3 (x+ 1 )+( 2 x− 1 )(x− 1 ) (x− 1 )(x+ 1 )^2 = 3 x+ 3 + 2 x^2 − 3 x+ 1 (x− 1 )(x+ 1 )^2 = 2 x^2 + 4 (x− 1 )(x+ 1 )^2 Here we h ...

If you really know your difference of two squares then you might spot a short cut here: 1 x− 1 + 1 x+ 1 − 1 x+ 2 = 2 x x^2 − 1 − ...

In this example division byx−1resultedinaremainderof3/(x− 1 ). In general, when a polynomialp(x)is divided by a linear factorx−a ...

2.2.10 Partial fractions ➤ 39 77 ➤ Often we have to reverse the process of taking the common denominator, i.e. split an expressi ...

and use the fact that this must be an identity to get relations betweenAandB: A+B= 1 −( 2 A+B)= 1 the solution of which reproduc ...

side is always the same as the total degree of the original denominator. In the above the cover-up rule may be used forA,B,Cin t ...

in any event it only requires care to use=and appropriately. In algebra we consider only exact solutions, always use=and avoid d ...

The problem with a quadratic equation occurs when it won’t factorise. But in all cases we can use the following formula for the ...

We will do a numerical example in parallel with the symbolic form as a concrete illustra- tion. We start with the general quadra ...

Sincea =0thisgives ( x+ b 2 a ) 2 = b^2 − 4 ac ( 2 a)^2 Hence x+ b 2 a =± √ b^2 − 4 ac 2 a so x=− b 2 a ± √ b^2 − 4 ac 2 a = −b ...