# 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