AS Level Pure Mathematics

• The laws of indices involve a base number, which can be represented as 'n', and something happening to the base number, which is often represented as 'a' or 'b'.
• The law states that if you're multiplying together the base number and a power, you can add the powers together.
• To solve a cubic equation, factorize it into three brackets that all multiply together, then rearrange to get a quadratic.
• To solve a line equation, combine two equations into one by making them equal to each other, then rearrange to get a quadratic.
• To solve an equation where a number term is missing on the left hand side, take 15 away from both sides to get it equal to zero, then factorize the quadratic to find the values of x and y.
• Integration is the process of turning a gradient into a line, and differentiation is the process of turning a line into a gradient.
• If you have cos(2x), then integrating it results in sin(2x).
• Differentiation and integration are the opposite processes, and they can be used to find the area between the curve and the x-axis.
• Trigonometry can be used in integration and differentiation, for example, if you have f(x) = cos(x), then integrating f(x) results in sine(x).
• An example of this law in action is 7 to the power of 2 multiplied by 7 to the power of 3, which can be simplified to 7 to the power of 5.
• Integrating increases all powers by 1 to the power of 3 becomes 1 to the power 4 and dividing by the new power 1 to the power 2.
• The first area's definite integral is calculated between the one and the four, resulting in 64.
• The second area's definite integral is calculated between the four and the six, resulting in 324.
• The definite integral is calculated for each cherry one at a time.
• There are also laws for subtraction, where the two indexes, or indices, are taken away.
• An example of this law in action is 3 to the power of 5 divided by 3 to the power of 2, which simplifies to 3 to the power of 3.
• The middle two terms of a quadratic will often cancel out to give zero.
• In a quadratic expansion, the numbers can be multiplied together first and then the letters can be multiplied together.
• When expanding brackets, it is important to multiply by both numbers as the common mistake is to only multiply by one of them.
• The numbers in the bracket can be multiplied by a negative version of the same bracket to expand the quadratic.
• In a linear expansion, the numbers can be multiplied together first and then the letters can be multiplied together.
• When the denominator of a fraction is a bracket with a certain number and a number inside, it cannot be multiplied through by root three.
• To solve a quadratic equation, multiply the first two terms and then multiply the result by the third term.
• The equation x² + 7x² = 25 is a quadratic equation.
• To figure out what x would be if a number equation is all equal to zero, make x equal zero or one of the brackets equal zero.
• To factorize a number, list the factors for the number and then jump around randomly until you find a pair of numbers which multiply together to make the number and add together to make a negative number.
• The equation x³ + 10x² = 25 is a cubic equation.
• To solve a cubic equation, start by multiplying the first two terms and then write in the third bracket and pretend it's a quadratic again.
• In algebraic long division, if the remainder is not zero, it means that the divisor is not a factor.
• In algebraic long division, the first term is divided by the divisor, and the remainder is calculated by multiplying the divisor by the remainder.
• In algebraic long division, if the remainder is zero, it means that the divisor is a factor.
• Algebraic long division can be used to factorize cubic expressions.
• Algebraic long division is used to divide expressions by a linear term.
• The third term in algebraic long division is divided by the new divisor, which is not the original divisor.
• Algebraic long division can be used to divide expressions into factors.
• The second term in algebraic long division is divided by the new divisor, which is not the original divisor.
• The process of algebraic long division continues until the remainder is zero.
• Perpendicular lines have a negative reciprocal gradient.
• The gradient of a line is the change in y over the change in x.
• The length of a line can be found using Pythagoras' theorem with the distance in x and y.