Polynomials Mind Map

A zero of a polynomial is the value of x for which the polynomial becomes 0. On the graph, it is the x-coordinate where the curve touches or cuts the x-axis.

When you see a graph, count the points where it meets the x-axis. Those x-values are the zeros.

A linear polynomial has highest power 1, usually written as ax+b where a is not zero. Its zero is the value of x that makes ax+b equal to 0.

For a linear polynomial ax+b, write the zero directly as -b/a after checking that a is not zero.

A quadratic polynomial is a polynomial of degree 2, usually written as ax^2+bx+c where a is not zero. Its zeros are the values of x that make the polynomial equal to 0.

If a quadratic factorises neatly, first break it into factors and then set each factor equal to zero.

A cubic polynomial is a polynomial of degree 3, usually written as ax^3+bx^2+cx+d where a is not zero. Its zeros are the values of x that make the polynomial equal to 0.

After factorising a cubic polynomial, keep solving until the expression is fully broken into linear factors.

If α and β are the zeros of the quadratic polynomial ax^2+bx+c, then α+β=-b/a and αβ=c/a, where a is not zero.

For ax^2+bx+c, always write sum as -b/a and product as c/a before substituting numbers.

If α, β, and γ are the zeros of ax^3+bx^2+cx+d, then α+β+γ=-b/a, αβ+βγ+γα=c/a, and αβγ=-d/a.

Write the cubic in standard form first, then apply all three relations one by one.

If α and β are the zeros of a quadratic polynomial, then the polynomial can be written as k(x-α)(x-β), where k is a non-zero constant.

From zeros α and β, use x-α and x-β, then expand carefully.

If α, β, and γ are the zeros of a cubic polynomial, then the polynomial can be written as k(x-α)(x-β)(x-γ), where k is a non-zero constant.

For three zeros, always write three factors before expanding. Do not skip any root.

To verify whether a number is a zero of a polynomial, substitute that number into the polynomial and check whether the value becomes 0.

Always write the substitution step clearly. One small sign mistake can change the final answer.

For polynomials p(x) and g(x), with g(x) not equal to zero, p(x)=g(x)q(x)+r(x), where the degree of r(x) is less than the degree of g(x).

Always check that the remainder has lower degree than the divisor before finalising the answer.

If a polynomial p(x) is divided by x-a, then the remainder is p(a). If p(a)=0, then x-a is a factor of the polynomial.

When the divisor is x-a, substitute a directly into p(x).

The sign pattern of a polynomial tells whether the graph lies above or below the x-axis in different intervals. The x-intercepts show the zeros, and the shape of the graph helps interpret the number of real zeros.

Read the graph from left to right and note where it is above, on, or below the x-axis.