Maths Methods 3/4 Polynomials Notes

Summary:

Polynomials are mathematical expressions of terms with variables raised to a whole number exponent and coefficients. These expressions come in various degrees, where the degree determines the highest power of the variable in the polynomial. Quadratic functions are a specific type of polynomial of degree 2, and they generally take the form �(�)=��2+��+�f(x)=ax2+bx+c. Several methods exist to solve quadratic equations, including factorizing, completing the square, and using the quadratic formula �=−�±�2−4��2�x=2a−b±b2−4ac​​. The discriminant, denoted as $\Delta$ and given by �2−4��b2−4ac, plays a vital role in determining the nature of the roots of the quadratic equation. If $\Delta >0$, the equation has two distinct solutions; for $\Delta =0$, there’s a single solution, and if $\Delta <0$, there are no real solutions.

The form of a parabola can be determined based on specific information. Given a turning point, one can use the form �=�(�−ℎ)2+�y=a(x−h)2+k. With x-intercepts, it takes the form $y=a(x-b)(x-c)$. If provided three points, one can use the general form �=��2+��+�y=ax2+bx+c and solve the equations simultaneously.

For higher-degree polynomials, the factor theorem and rational root theorem can assist in finding integer or rational roots, respectively. Once a polynomial is factorized, solving for its roots becomes simpler. Dividing polynomials can be done through long or synthetic division methods, both of which provide the quotient of the division.

Excerpt:

Maths Methods 3/4 Polynomials Notes

Polynomials
Polynomials are functions of the form, where is the degree of the polynomial is the leading coefficient and is the constant

Solving polynomial equations: Take all terms to one side and factorise, then apply the null factor law

Example qs:

1. Solve each of the following for x

4 = x3 + 3×2
0 = 8×3 + 27
x3 + 3×2 = 6x + 8
2×3 − x2 − 6x + 3 = 0