SHS Mathematics

Wassce 2019 Elective/Further Mathematics Paper Two Questions 5

Wassce 2019 Elective/Further Mathematics Paper Two Questions 7

Wassce 2021 Elective/Further Mathematics Paper Two Questions 3(a)

Wassce 2021 Core/General Mathematics Paper Two Questions 2(b)

Wassce 2021 Core/General Mathematics Paper Two Questions 2(a)

Wassce 2021 Elective/Further Mathematics Paper Two Questions 2

Wassce 2021 Elective/Further Mathematics Paper Two Questions 1

Hello, everyone.

Please, let's all note that from now till August,2022 I will be providing solutions to some mathematics past questions (both elective and Core maths) and our focus will be on the last 5-years(2016 to 2021).

Do yourself a favor by asking questions should you have any challenge with the solution that will be provided for clarification. We're here to learn; nothing much, nothing less.

Stay tuned, learn for yourself and thank yourself later

Get in touch:
https://bit.ly/3zKZazz
https://bit.ly/3MW9VSI

Stay Safe...✊

Solution to the question


maths

Suggested Solution To The Question Of The Day

QUESTION OF THE DAY

A man deposited an amount of money in his savings account for 5 years. The rate of interest is 14% per annum. If the interest was $35,000.00
Find the amount deposited.

Which is greater?

Solution...📝📝📐

Can you solve this question?

what are your answers?😊😊

How to Factor a Polynomial Expression

In mathematics, factorization or factoring is the breaking apart of a polynomial into a product of other smaller polynomials. If you choose, you could then multiply these factors together, and you should get the original polynomial (this is a great way to check yourself on your factoring skills).

One set of factors, for example, of 24 is 6 and 4 because 6 times 4 = 24. When you have a polynomial, one way of solving it is to factor it into the product of two binomials.

You have multiple factoring options to choose from when solving polynomial equations:

For a polynomial, no matter how many terms it has, always check for a greatest common factor (GCF) first. Literally, the greatest common factor is the biggest expression that will go into all of the terms. Using the GCF is like doing the distributive property backward.

If the equation is a trinomial — it has three terms — you can use the FOIL method for multiplying binomials backward.

If it’s a binomial, look for difference of squares, difference of cubes, or sum of cubes.

Finally, after the polynomial is fully factored, you can use the zero product property to solve the equation.

To be continued...

source: unknown

Identifying Polynomial Functions

We have introduced polynomials and functions, so now we will combine these ideas to describe polynomial functions. Polynomials are algebraic expressions that are created by adding or subtracting monomial terms, such as
−5x^2
, where the exponents are only integers. Functions are a specific type of relation in which each input value has one and only one output value.

Polynomial functions have all of these characteristics as well as a domain and range, and corresponding graphs. In this section, we will identify and evaluate polynomial functions. Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the powers of the variables.

When we introduced polynomials, we presented the following:
4x^3 −9x^2 + 6x. We can turn this into a polynomial function by using function notation:
f(x)=4x^3 − 9x^2 + 6x

Polynomial functions are written with the leading term first and all other terms in descending order as a matter of convention.

To be continued...

Polynomial Questions With Answers

Question 1
Write polynomial P as a product of linear factors : P(x) = x² - 4.

Question 2
Write polynomial P as a product of linear factors : P(x) = 9x³ - 19x - 10.

Question 3
Write polynomial P as a product of linear factors : P(x) = 5x³ -2x² + 5x - 2

Question 4
Write polynomial P as a product of linear factors : P(x) = -x³ -2x² + 5x + 6

Question 5
Write polynomial P as a product of linear factors : P(x) = (x + 2)(20x² + 41x + 20)

Question 6
Show that 1 and -3 are zeros of polynomial P given by: P(x) = x⁴ + 4x³ + 6x² + 4x - 15

Question 7
Show that -1 and -5 are zero of polynomial P : P(x) = x⁴ + 12x³ + 54x²+ 108x + 65

Question 8
Show that -2, -3 and 2 are zeros of polynomial P : P(x) = x⁵ + x⁴ + 26x² -16x -120

ANSWERS TO ABOVE QUESTIONS
1) P(x) = (x - 2)(x + 2)
2) P(x) = (x + 1)(3x + 2)(3x - 5)
3) P(x) = (x + i)(x - i)(5x - 2)
4) P(x) = (x + 1)(2 - x)(x + 3)
5) P(x) = 20(x + 2)(x + 5/4)(x + 4/5)
6) P(1) = 0 , P(-3) = 0

7) P(- 1) = 0 , P(- 5) = 0

8) P(- 2) = 0 , P(-3) = 0 , P(2) = 0


To be continued

2. Examples of polynomial functions

Here are some examples of polynomial functions and the language we use to describe them:
f(x)=3x−2 Linear polynomial (linear function)
f(x)=x²−4x+1 Quadratic polynomial
f(x)=−3x²+x−6 Cubic polynomial with no quadratic term
f(x)=(x−3)²(2x−1) Cubic polynomial (convince yourself that the largest power will be three when expanded)
f(x)=2x⁴−5x³+2x³−x+17 Quartic polynomial
f(x)=18x⁵−7 Quintic polynomial with only the 5th degree and constant terms.

To be continued...

1. Polynomial

Definitions & examples
Polynomial functions are functions of a single independent variable, in which that variable can appear more than once, raised to any integer power. For example, the function

f(x)=8x⁴−4x³+3x²−2x+22
is a polynomial. Polynomial functions are sums of terms consisting of a numerical coefficient multiplied by a unique power of the independent variable.

We generally write these terms in decreasing order of the power of the variable, from left to right*. Here is a summary of the structure and nomenclature of a polynomial function:

*Note: There is another approach that writes the terms in order of increasing order of the power of x. This has some appeal because we write power series that way. You'll have to choose which works for you.

To be continued...

4. Formulas
The general algebraic formulas we use to solve the expressions or equations are:

(a + b)²= a² + 2ab + b²
(a – b)² = a² – 2ab + b²
a² – b² = (a – b)(a + b)
(a + b)³ = a³ + b³ + 3ab(a + b)
(a – b)³ = a³ – b³ – 3ab(a – b)
a³ – b³ = (a – b)(a² + ab + b²)
a³ + b³ = (a + b)(a² – ab + b²)

Solved Problem
Example: Simplify the given expressions by combining the like terms and write the type of Algebraic expression.
(i) 3xy­­³ + 9x² y³ + 5y³x

(ii) 7ab² c² + 2a³ b² − 3abc – 5ab² c² – 2b² a³ + 2ab

(iii) 50x³ – 20x + 8x + 21x³ – 3x + 15x – 41x³

To be continued...

3. We may or might have come across the terms of algebraic equations such as:

Coefficient of a term
Variables
Constant
Factors of a term
Terms of equations
Like and Unlike terms
Example of using these terms are given below.

If 2x2+3xy+4x+7 is an algebraic expression.

Then, 2x2, 3xy, 4x and 7 are the terms

Coefficient of term x2 = 2

Constant term = 7

Example of like and unlike terms:

Like terms: 2x and 3x

Unlike terms: 2x and 3y

Factors of a term:

If 3xy is a term, then its factors are 3, x and y.

Monomial, Binomial & Trinomial

Also, in grade 7 we will learn about types of expressions, such as monomial, binomial and trinomial. Let us see examples of each.

Monomial: 2x

Binomial: 2x+3y

Trinomial: 2x+3y+9

Addition and Subtraction of Algebraic Expressions

We can add and subtract like terms easily.

Example: Add 3x + 5y – 6z and x – 4y + 2z.

By adding both the expressions we get;

(3x + 5y – 6z) + (x – 4y + 2z)

Separating the like terms and adding them together:

(3x + x) + (5y – 4y) + (-6z + 2z)

4x + y – 4z
To be continued...

2.Types of Algebraic expression
There are 3 main types of algebraic expressions which include:

Monomial Expression
Binomial Expression
Polynomial Expression
Monomial Expression

An algebraic expression which is having only one term is known as a monomial.

Examples of monomial expression include 3x4, 3xy, 3x, 8y, etc.

Binomial Expression
A binomial expression is an algebraic expression which is having two terms, which are unlike.

Examples of binomial include 5xy + 8, xyz + x3, etc.

Polynomial Expression
In general, an expression with more than one terms with non-negative integral exponents of a variable is known as a polynomial.

Examples of polynomial expression include ax + by + ca, x3 + 2x + 3, etc.

Other Types of Expression:
Apart from monomial, binomial and polynomial types of expressions, an algebraic expression can also be classified into two additional types which are:

Numeric Expression
Variable Expression
Numeric Expression

A numeric expression consists of numbers and operations, but never include any variable. Some of the examples of numeric expressions are 10 + 5, 15 ÷ 2, etc.

Variable Expression
A variable expression is an expression which contains variables along with numbers and operation to define an expression. A few examples of a variable expression include 4x + y, 5ab + 33, etc.
To be continued...

What is an Algebraic Expression?

An algebraic expression in mathematics is an expression which is made up of variables(x, y, z etc...) and constants(a, b, c or 2, 3 6, 1etc) along with algebraic operations (addition, subtraction, etc.). Expressions are made up of terms.

Examples
3x + 4y – 7, 4x – 10, etc.

These expressions are represented with the help of unknown variables, constants and coefficients. The combination of these three (as terms) is said to be an expression. It is to be noted that, unlike the algebraic equation, an algebraic expression has no sides or equal to sign. Some of its examples include

3x + 2y – 5
x – 20
2x2 − 3xy + 5

Variables, Coefficient & Constant
In Algebra we work with Variable, Symbols or Letters whose value is unknown to us.

Algebraic Expression

In the above expression (i.e. 5x – 3),

x is a variable, whose value is unknown to us which can take any value.
5 is known as the coefficient of x, as it is a constant value used with the variable term and is well defined.
3 is the constant value term which has a definite value.
The whole expression is known to be the Binomial term, as it has two unlikely terms.
To be continued...