Polynomial Equation – HSC Math

Posted In: Mathematics

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate $x$ is $x^2−4x+7$. An example in three variables is $x^3+2xyz^2−yz+1$.

  1. Nature of roots of a quadratic equation by its discriminant $b^2-4ac$
    $b^2-4ac>0,$ Roots are unequal and irrational.
    $b^2-4ac=0,$ Roots are equal and real.
    $b^2-4ac<0,$ Roots are unequal and complex. $b^2-4ac>0$ is perfect square, Roots are unequal and rational.
  2. if $α$ and $β$ are roots of a quadratic equation $ax^2+bx+c=0 then,$
    i. Sum of roots, $α+β=-b/a$
    i. Product of roots, $αβ=c/a$
  3. $ax^2+bx+c=0,\ \where\ a,b,c∈R$
    i. if $c=0,$ one of the root is zero.
    ii. if $b=0,$ absolute values of roots are equal.
    iii. if $a$ and $c$ are of opposite sign then the roots are real.
    iv. if Discriminant, $D=b^2-4ac=0$ then, $ax^2+bx+c$ is a perfect square.
  4. For cubic equation $ax^3+bx^2+cx+d=0$
    i. $∑α=α+β+γ=-b/a$
    ii. $∑αβ=αβ+βγ+γα=c/a$
    iii. $∑αβγ=αβγ=-d/a$
  5. For Quartic equation $ax^4+bx^3+cx^2+dx+e=0$
    i. $∑α=α+β+γ+δ=-b/a$
    ii. $∑αβ=αβ+βγ+γδ+δα=c/a$
    iii. $∑αβγ=αβγ+βγδ+γδα+αβδ=-d/a$
    iv. $∑αβγδ=αβγδ=e/a$
  6. If the roots of an equation follow Arithmetic Progression (A.P.)-
    i. for cubic equation, roots are $a-d, a, a+d$
    ii. for quartic equation, roots are $a-3d, a-d, a+d, a+3d$
  7. If the roots of an equation follow Geometric Progression (A.P.)-
    i. for cubic equation, roots are considered $a/r,a,ar$
    ii. for quartic equation, roots are $a/r^3,a/r,ar,ar^3$

In mathematics, an arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant. For example, the sequence 1, 2, 3, 4, … is an arithmetic progression with common difference 1. Similarly the sequence 3, 5, 7, 9, 11,… is an arithmetic progression
with common difference 2.

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, … is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, … is a geometric sequence with common ratio 1/2.


Post Tags: Algebra | HSC


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